Notes on Representations of Real Reductive Groups Math 224, Spring 2017

NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS MATH 224, SPRING 2017

DENNIS GAITSGORY

1. Week 1, Day 1 (Tue, Jan 24) 1.1. Action of distributions.

1.1.1. Let K be a compact topological group. Let C(K) denote the space of continuous complex- valued functions on K. Unless speciﬁed otherwise, integration over K will be with respect to the Haar measure µHaar on K, normalized so that the volume of K equals 1. Let Meas(K) be the topological dual of C(K) (it is known that Meas(K) can be described as Radon measures). Unless speciﬁed otherwise, we will regard it as equipped with the weak topology.

For k ∈ K we let δk ∈ Meas(K) denote the corresponding δ-function, i.e., hδk, fi = f(k). Note that we have a canonical embedding

(1.1) L1(K) ,→ Meas(K), f 7→ f · µHaar with dense image (with respect to the weak topology on Meas(K)). Remark 1.1.2. Note that, being the topological dual of a Banach space, Meas(K) has a natural norm, isn which it is a Banach space. The corresponding topology (called the norm topology) is stronger than the weak topology. The map (1.1) is an isometric embedding with respect to the corresponding norms (in particular, the image of L1(K) in Meas(K) is not dense in the norm topology).

The above discussion is applicable to (K, µHaar) replaced by an arbitrary compact topological set equipped with a Borel measure.

1.1.3. Pushforward of measures along the multiplication map K × K → K (i.e., the operator dual to that of pullback of functions) makes Meas(K) into an associative algebra. We denote the corresponding operation by

f1, f2 7→ f1 ? f2 and refer to it as the convolution product. The unit for this algebra is δ1. We have

δk1 · δk2 = δk1·k2 .

This algebra structure is compatible with the embedding (1.1), i.e., it induces a structure of associative algebra on L1(K). We have

µHaar ? µHaar = µHaar.

Date: September 22, 2021. 1 2 DENNIS GAITSGORY

1.1.4. Let V be a ﬁnite-dimensional continuous representation of K. For k ∈ K we let Tk the corresponding element of End(V ). Since for every v ∈ V, v∗ ∈ V ∗, the matrix coeﬃcient function

∗ k 7→ hv ,Tk(v)i is continuous, we obtain that the assignment k 7→ Tk uniquely extends to a continuous linear map

(1.2) Meas(K) → End(V ), f 7→ Tf , determined by the condition that Tδk = Tk. The map (1.2) is a homomorphism of associative algebras.

1.2. Matrix coeﬃcients.

1.2.1. We view C(K) as a representation of K × K by

−1 ((k1, k2) · f)(k) = f(k1 · k · k2).

By adjunction, we obtain action of K × K on Meas(K). We have

(k1, k2) · δk = δ −1 . k1·k·k2

1.2.2. For a ﬁnite-dimensional continuous representation V of K, we view End(V ) as a representation of K × K via

−1 (k1, k2) · S = Tk1 ◦ S ◦ T . k2

The action map

(1.3) ActV : Meas(K) → End(V ), f 7→ Tf and the matrix coeﬃcient map

(1.4) MCV : End(V ) → C(K), MCS,V (k) = Tr(S ◦ Tk−1 ,V ) are maps of K × K-representations.

1.2.3. Note that we can canonically identify End(V ) with the dual of End(V ∗) as K × K- representations by setting

∗ ∗ hS1,S2i = Tr(S1 ◦ S2 ,V ),S1 ∈ End(V ),S2 ∈ End(V ), where S 7→ S∗ denotes the operation of taking the dual linear operator. In terms of this identiﬁcation, the maps (1.3) and (1.4) are each other’s duals. Remark 1.2.4. Note that the map S 7→ S∗ deﬁnes an isomorphism of vector spaces End(V ) ' End(V ∗). This isomorphism is compatible with the K × K-actions up to the swap of factors in K × K. 1.3. Orthogonality formulas. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 3

1.3.1. Let U be a ﬁnite-dimensional continuousrepresentation of K. Note that the operator inv K PU = TµHaar is an idempotent projection onto U , the subspace of K-invariant vectors. For S ∈ End(U) we have Z inv inv inv K (1.5) MCS,U (k) = Tr(S ◦ PU ,U) = Tr(PU ◦ S ◦ PU ,U ). k∈K

Proposition 1.3.2. Let V1 and V2 be ﬁnite-dimensional continuous irreducible representations. Then we have: Z

MCS1,V1 (k) · MCS2,V2 (k) = 0 k∈K ∗ ∗ unless V2 ' V1 (both reps are irreducible), and for V1 = V and V2 = V , we have Z 1 ∗ MC (k) · MC ∗ (k) = · Tr(S ◦ S ,V ). S1,V S2,V dim(V ) 1 2 k∈K

∗ Proof. Consider U = V1 ⊗ V2, and apply (1.5) to S = S1 ⊗ S2. If V2 is non-isomorphic to V1 , then U K = 0, and the assertion follows.

∗ If V1 = V and V2 = V , we identify V ⊗ V ∗ ' End(V ),

∗ and under this identiﬁcation, for S1 ∈ End(V ) and S2 ∈ End(V ), the corresponding endomor- ∗ phism S1 ⊗ S2 of V ⊗ V corresponds to the endomorphism of End(V ) given by

0 0 ∗ (1.6) S 7→ S1 ◦ S ◦ S2 .

By Schur’s lemma, the map

K C → End(V ) , 1 7→ IdV is an isomorphism. The corresponding projector

inv K PEnd(V ) : End(V ) → End(V ) identiﬁes with S0 7→ Tr(S0,V ).

The operator 0 inv 0 inv S 7→ PEnd(V ) ◦ S ◦ PEnd(V ), End(End(V )) → C sends the endomorphism of End(V ) given by (1.6) to 1 · Tr(S ◦ S∗,V ). dim(V ) 1 2 This establishes the desired identity. 4 DENNIS GAITSGORY

1.3.3. Note that −1 MCS∗,V ∗ (k) = MCS,V (k ). Hence, the orthogonality relation can also be interpreted as a formula for Z −1 (1.7) MCS1,V1 (k) · MCS2,V2 (k ). k∈K

Namely, (1.7) vanishes unless V1 ' V2, and for V1 = V = V2, it equals 1 · Tr(S ◦ S ,V ). dim(V ) 1 2 1.3.4. As a corollary of Proposition 1.3.2, we obtain:

Corollary 1.3.5. For V1 and V2 irreducible, the composite

ActV1 MCV2 End(V1) −→ C(K) ,→ Meas(K) −→ End(V2) 1 is zero unless V1 ' V2 and equals dim(V ) · IdEnd(V ) for V1 = V = V2. 1.3.6. Using an invariant Hermitian form, we also have −1 MCS,V (k ) = MCS†,V (k).

Thus, we obtain that the images of MCV for distinct V ’s are pair-wise orthogonal in L2(K), and that for an individual V , the map MCV is an isometric embedding of End(V ) into L2(K), where we endow End(V ) with a Hermitian structure by the formula 1 (1.8) (S ,S ) = · Tr(S ◦ S†). 1 2 dim(V ) 1 2 1.4. Characters.

1.4.1. Let V be a ﬁnite-dimensional continuous representation of K. Let χV ∈ C(K) be equal MCV (IdV ). From Proposition 1.3.2 we obtain: Corollary 1.4.2. Let V and W be irreducible. Then Z Z Z −1 χV (k) · χW (k ) = χV (k) · χW ∗ (k) = χV (k) · χW (k) k∈K k∈K k∈K equals 1 if V ' W , and vanishes otherwise.

1.4.3. For V irreducible, set ξV := dim(V ) · χV . From Corollary 1.3.5, we obtain:

Corollary 1.4.4. For W irreducible, the element TξV ∈ End(W ) equals IdW for W ' V and zero otherwise. From Corollary 1.7, we obtain:

Corollary 1.4.5. ξV ? ξW = 0 unless W ' V and ξV ? ξV = ξV .

2. Week 1, Day 2 (Tue, Jan 26) 2.1. Continuous representations. 2.1.1. Let G be a topological group and V a topological vector space. We shall say that a representation of G on V is continuous if the action map G × V → V is continuous. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 5

2.1.2. In this course we will restrict our attention to the following class of topological vector spaces: we will assume that they are Hausdorﬀ, 2nd countable, locally convex and complete.

Locally convex is equivalent to saying that there is a fundamental system of neighborhoods of 0 ∈ V of the form {v ∈ V , ρ(v) < 1}, where ρ : V → R is a continuous map satisfying

ρ(c · v) = |c| · ρ(v), ρ(v1 + v2) ≤ ρ(v1) + ρ(v2).

Such maps are called semi-norms.

2.1.3. Suppose that G is locally compact. Here is a (tautological) way to reformulate the condition of continuity of action. It amounts to the combination of the following two:

(i) For every v ∈ V , a semi-norm ρ and , there exists a neighborhood 1 ∈ U ⊂ G such that

ρ(Tg(v) − v) < for g ∈ U.

(ii) For every compact Ω ⊂ G and every semi-norm ρ, there exists a semi-norm ρ0 so that

0 ρ(Tg(v)) ≤ ρ (v), for all v ∈ V and g ∈ Ω.

2.1.4. Here are some examples of continuous representations. Let G act continuously on a topological space X.

(a) Let C(X) be the space of continuous functions on G. We topologize by convergence on compact subsets. I.e., the fundamental system of neighborhoods of 0 is given by semi-norms

max |f(x)|, x∈Ω where Ω is a compact subset of X.

We consider the action of G on V = C(X) by

−1 (Tg · f)(x) = f(g · x).

Conditions (i) and (ii) are evident in this case.

(b) Let µ be a (positive) measure on X. I.e., consider Cc(X) equipped with the sup norm, µ is a continuous functional. Assume that G acts on µ continuously: i.e., for every compact Ω ⊂ G there exists a scalar c ∈ R+ such that g(µ) ≤ c · µ.

Take V = Cc(X), but equip it with the topology induced by the Lp norm with respect to µ. Conditions (i) and (ii) are again evident.

(b’) Take V = Lp(X, µ), i.e., the completion of the space of from the previous example in its norm. Conditions (i) and (ii) follow formally from the fact that they hold on a dense subset. 6 DENNIS GAITSGORY

2.1.5. Let V be a topological vector space as in Sect. 2.1.2. Let X be a topological space, and let f : X → V be a continuous function. Proposition 2.1.6. The assignment

δx 7→ f(x) uniquely extends to a continuous linear map Z Measc(X) → V, µ 7→ f µ ∗ where Measc(X) = C(X) , equipped with the weak topology. 0 Proof. Note that the linear span of the elements δx, denoted Measc (X), is dense in Measc(X), in the weak topology. By the completeness of V , we need to show that the map Z 0 µ 7→ f, Measc (X) → V µ0 0 is continuous (in the topology on Measc (X), induced by the weak topology on Measc(X)). This is equivalent to showing that for every semi-norm ρ on V , there exists a neighborhood 0 0 0 ∈ U ⊂ Measc (X), such that Z ρ( f) < 1 for µ0 ∈ U 0.

µ0 However, we have Z ρ( f) ≤ h|µ0|, ρ(f)i,

µ0 by convexity. 0 Hence, we can take U to correspond to the neighborhood U ⊂ Measc(X) given by {µ , |h|µ|, ρ(f)i| < 1}.

2.1.7. As a corollary, we obtain that for a continuous representation V of G and v, we have a well-deﬁned linear map

µ 7→ µ ? v : Measc(G) → V, which is continuous in the weak topology on Measc(G), and such that δg ? v = Tg(v). Moreover, it is easy to see that the resulting map

Measc(G) × V → V, (µ, v) 7→ µ ? v is continuous. For a ﬁxed µ, we will denote the corresponding endomorphism of V by Tµ, and the map

Measc(G) → End(V ) by ActV ; it is continuous at least for the weak topology on End(V ). NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 7

2.1.8. By a Dirac sequence we will mean a family of continuous compactly supported functions fn on G such that fn · µHaar → δ1 in the weak topology on Measc(G). The latter condition R follows from the following two: fn → 1 and supp(fn) → {1}. We obtain that for a continuous representation V and v ∈ V ,

(2.1) fn ? v → v.

2.2. The notion of K-ﬁnite vector.

2.2.1. Let K be compact, and V a continuous representation of K. We let V K -ﬁn denote the subspace of V equal to the union of ﬁnite-dimensional K-subrepresentations of V .

K -ﬁn Clearly, an element v ∈ V belongs to V if and only if the elements Tk(v) span a ﬁnite- dimensional subspace of V .

2.2.2. For an irreducible ﬁnite-dimensional representation ρ, we let

V ρ ⊂ V K -ﬁn be the ρ-isotypic component, i.e., the union of ﬁnite-dimensional subrepresentations that are isomorphic to direct sums of copies of ρ.

Clearly, we have ρ V ' ρ ⊗ HomK (ρ, V ), where HomK (ρ, V ) ⊂ Hom(ρ, V ) inherits a topology from V . We have V K -ﬁn = ⊕ V ρ. ρ∈Irrep(K)

2.2.3. We claim:

Lemma 2.2.4. Let ρ be an irreducible ﬁnite-dimensional representation. Then for any S ∈ End(ρ∗), the image

MCS,ρ ·µHaar ∈ Meas(K) acting on V belongs to V ρ.

Proof. Follows from the fact that

δk ? (MCS,ρ ·µ) = MCTg ·S,ρ .

Corollary 2.2.5. For an irreducible representation ρ, the element ξρ · µHaar ∈ Meas(K) acts in any continuous representation V as an idempotent with image equal to V ρ.

Proof. Follows from Lemma 2.2.4 and Corollary 1.4.4.

2.3. Peter-Weyl theorem. 8 DENNIS GAITSGORY

2.3.1. Recall that we have an isometric embedding

(2.2) ⊕ˆ End(ρ) → L2(K), ρ∈Irrep(K) where each End(ρ) is endowed with a Hermitian structure by formula (1.8). The Peter-Weyl theorem says: Theorem 2.3.2. (a) The map is an isomorphism.

(b) The subspace ⊕ End(ρ) ⊂ L2(K) identifies with the set of K-finite vectors with ρ∈Irrep(K) respect to the action by left translations. 2.3.3. We will first prove point (b). This consists of the following two statements:

ρ,l Proposition 2.3.4. The map MCρ : End(ρ) → C(K) is an isomorphism onto C(K) , where the superscript l indicates that we are considering C(K) as equipped with the action by left translations. Proof. This is just Frobenius reciprocity: ∗ HomK (ρ, C(K)) ' ρ . K -ﬁn,l K -ﬁn,l Proposition 2.3.5. The inclusion C(K) ,→ L2(K) is an isomorphism. Proof. We will need the following lemma:

Lemma 2.3.6. Let W be a ﬁnite-dimensional representation of K. Then IdW lies in the image of C(K) ⊂ Meas(K) under ActW .

Proof. On the one hand, the closure of the image of C(K) contains IdW by (2.1). On the other hand, the said image is a linear subspace of End(W ), and hence is closed.

Let f be a (left) K-ﬁnite element in L2(K). By the above lemma, we can ﬁnd f1 ∈ C(K) so that f1 ? f = f (note that the operator Tf1 for V = L2(K) amounts to the convolution f1 ? −). However, as we shall see below (in Sect. 2.4.3), for any f1 ∈ C(K) and f ∈ L1(K), the convolution f1 ? f belongs to C(K). 2.4. Proof of density.

2.4.1. To prove point (a) of Theorem 2.3.2, we need to show that for any v ∈ L2(K) and there 0 K -ﬁn,l 0 exists v ∈ L2(K) such that ||v − v || < .

We recall that an operator T : V1 → V2 between Banach spaces is called compact if the closure of the image of the unit ball in V1 is compact in V2. We will use the following key observation: Theorem 2.4.2. Let X and Y be compact metric spaces, and let F (−, −) be a continuous function on X × Y . Then for µ ∈ Meas(X), the function Z Tµ,F (y) = F (x, y) x∈X,µ NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 9 is continuous. Moreover, the resulting operator

TF : Meas(X) → C(Y ), is compact, when Meas(X) is considered as equipped with its natural norm.

Proof. We will show that Tµ,F is bounded and uniformly continuous with the estimates depending only on ||µ||. This will prove the theorem in view of the Arzel`a-Ascolitheorem. Indeed, for boundedness we note that for any y ∈ Y

|Tµ,F (y)| ≤ ||µ|| · sup(F ).

For uniform continuity ﬁx an . Since F is uniformly continuous, we can ﬁnd a δ such that ρ(y , y ) < δ ⇒ |F (x, y ) − F (x, y )| < ∀x ∈ X. 1 2 1 2 ||µ|| However, this implies that

|Tµ,F (y1) − Tµ,F (y2)| < . −1 2.4.3. We apply the above observation to X = Y = K, and F (k1, k2) = f(k1 · k2), where f ∈ C(K) We note that the corresponding operator

TF : Meas(K) → C(K) equals µ 7→ µ ? f.

In particular, we obtain that the latter operator is compact. Hence, the operator r Tf : L2(K) → L2(K), being equal to the composition

µ7→µ?f L2(K) → Meas(K) −→ C(K) → L2(K), is also compact.

2.4.4. For a given v ∈ L2(K), let us choose f to be a continuous function on K such that ||v − v ? f|| < 2 ; it exists by (2.1), where we regard L2(K) as equipped with the action on K by right translations. We can furthermore assume that f is such that f is real-valued and satisfies f(g) = f(g−1). r In this case the operator Tf : L2(K) → L2(K) is self-adjoint. We now quote the following theorem (which is proved in the same way as its finite-dimensional counterpart): Theorem 2.4.5. Let T be a compact self-adjoint operator on a Hilbert space H. Then H admits a direct sum decomposition Mˆ H ' H0 ⊕ Hλ, λ where Hλ is the λ-eigenspace of T . Moreover: (i) for λ 6= 0 the space Hλ is finite-dimensional; (ii) for every all but finitely many λ’s satisfy |λ| < . 10 DENNIS GAITSGORY

r We apply this theorem for H = L2(K) and T = Tf . Note that since left and right multipli- cations commute, we obtain that each Hλ is left K-invariant. In particular, we obtain that for λ K -fin,l λ 6= 0, we have H ⊂ L2(K) . λ λ λ Decompose v as v0 + Σ v with each v ∈ H . Note that λ6=0 λ v − v ? f = v0 + Σ (1 − λ) · v . λ6=0 Hence, ||v − v ? f|| < ⇒ ||v − Σ vλ|| < . 2 |λ|> 2 Thus, the (finite) sum Σ vλ gives the desired approximation to v by left K-finite vectors. |λ|> 2 This finishes the proof of Theorem 2.3.2(a). 2.5. Density in other representations. We will now prove: Theorem 2.5.1. For any continuous representation V of K, the subset V K -fin ⊂ V is dense.

Proof. Let v be a vector in V . Choose a Dirac sequence pf continuous functions fn so that

fn ? v → v. 0 0 1 By Peter-Weyl, we can choose K-finite functions fn such that ||fn − fn||L2 < n . Then the 0 sequence fn also converges to δ1 in the weak topology on Meas(K). Hence, 0 fn ? v → v. 0 However, each fn ? v is K-finite by Lemma 2.2.4. Corollary 2.5.2. Matrix coefficients of finite-dimensional representations are dense in the max norm in C(K). 2.5.3. As another corollary we obtain: Corollary 2.5.4. Any irreducible continuous representation of K is finite-dimensional. Proof. Let V be an irreducible representation. Recall that in the infinite-dimensional setting, “irreducible” means that V does not contain closed proper non-zero K-invariant subspaces. Since V K -fin is dense in V , it is non-zero. Let V 0 be an irreducible suprepresentation of V K -fin; it is tautologically finite-dimensional. Hence, V 0 is closed in V . Hence V 0 = V . 2.6. Plancherel’s formula. 2.6.1. Recall that the Peter-Weyl theorem says that the map

⊕ˆ End(V ) → L2(K) V ∈Irrep(K) (where ⊕ˆ means the Hilbert space direct sum) is an isomorphism. Since we already know that it is an isometric embedding, the assertion amounts to the fact that the above map has a dense image. It follows from Corollary 1.3.5 that the inverse map

L2(K) → ⊕ˆ End(V ), V ∈Irrep(K) NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 11 sends

(2.3) f 7→ ⊕ˆ dim(V ) · Tf . In particular, we have:

Theorem 2.6.2. For f1, f2 ∈ L2(K), we have Z X (2.4) f (k) · f (k) = dim(V ) · Tr(T ◦ T † ,V ), 1 2 f1 f2 k∈K V ∈Irrep(K) where the series on the right is absolutely convergent. The latter identity is one form Plancherel’s formula for compact groups. 2.6.3. Here is another identity (also sometimes referred to as Plancherel’s formula): Theorem 2.6.4. For a smooth function f on K, we have

(2.5) f(1) = Σ dim(ρ) · Tr(Tf , ρ), ρ∈Irrep(K) where the series on the right is an absolutely convergent. In order to prove this assertion, one need to recall the notion of trace class endomorphism of a Hilbert space. By deﬁnition, a compact endomorphism T : H → H is said to be of trace class if for some choice of orthonormal bases ei, the series

(|T |(ei), ei)

† 1 is absolutely convergent, where |T | = (T · T ) 2 . If this happens, one (easily) shows that the above condition and the resulting sum

Tr(T, H) := (T (ei), ei)

Lˆ does not depend on the choice of a basis. Moreover, if H is represented as Hi, then i

(2.6) Tr(T, H) = Σ Tr(Ti, Hi), i where Ti is the orthogonal projection of T onto Hi (part of the statement is that each Ti is trace class and the above series is absolutely convergent). Let us be in the situation of Theorem 2.4.2 with X = Y . Let X be equipped with a Borel measure µ; consider the corresponding embedding f7→f·µ C(K) −→ L2(K, µ) and its dual L2(K, µ) ,→ Meas(K, µ). For a continuous function F on X × X consider the corresponding composite (also abusively denoted TF ): TF L2(K, µ) ,→ Meas(K) −→ C(K) ,→ L2(K, µ). Proposition 2.6.5. Assume that X is a smooth manifold, and that µ and F are smooth. Under the above circumstances, TF is trace class and Z Tr(TF ,L2(X, µ)) = F (x, x). x∈X,µ 12 DENNIS GAITSGORY

−1 Proof of Theorem 2.6.4. We apply Proposition 2.6.5 to X = K and F (k1, k2) = f(k1 · k2 ), so l l that TF = Tf . By Proposition 2.6.5, the operator Tf is trace class and its trace equals f(1). l However, by (2.3) and (2.6), the RHS in (2.5) also equals Tr(Tf ,L2(K)).

3. Week 2, Day 1 (Tue, Jan 31) 3.1. Some diﬀerential calculus.

3.1.1. Let X be a differentiable manifold, and let V be a topological vector space as in Sect. 2.1.2. We shall say that a function F : X → V is differentiable at x, of there exists a linear map dFx : TxX → V such that for every vector ξx ∈ TxX and some/any differentiable path

γ :(−1, 1) → X, γ(0) = x, dγ0 = ξx we have F (γ(t)) − F (x) → dF (ξ ). t x x 1 n 1 n Let ξ , ..., ξ be a vector fields on X such that ξx, ..., ξx form a basis of TxX for every x ∈ X (such a frame of vector field exists locally on X). We shall say that F is continuously differentiable if the functions i Lieξi (F ); x 7→ dFx(ξx), i = 1, ..., n are continuous. This definition is easily seen to be independent of the choice of the frame. We let C1(X,V ) ⊂ C(X,V ) be the subspace of continuously differentiable functions. We define the spaces Cm(X,V ) inductively, by setting m m−1 m−1 C (X,V ) = {F ∈ C (X,V ) , Lieξi (F ) ∈ C (X,V ) ∀i = 1, ..., n}.

Set C∞(X,V ) = ∩ Cm(X,V ). m 3.1.2. We recall the notion of the ring of differential operators on X. Namely, we define ≤n ∞ Diff (X) ⊂ EndC(C (X)) inductively as follows: (i) Diff≤n(X) = 0 for n < 0; (ii) T ∈ Diff≤n(X) if and only if [T, f] ∈ Diff≤n−1(X) = 0 for every f ∈ C∞(X), viewed as an endomorphism g 7→ f · g. Clearly, Diff(X) := ∪ Diff≤n(X) n ∞ n ∞ is a subring of EndC(C (X)). Each Diff (X) is a bimodule over C (X). Thus, Diff≤0(X) = C∞(X). One can show that Diff≤1(X) = C∞(X) ⊕ Vect(X). Moreover, for a choice of a frame ξ1, ..., ξn as above, every differential operator can be uniquely written as

i1 in ∞ Σ fi1,...,in · ξ1 · ... · ξn , fi1,...,in ∈ C (X). i1,...,in≥0 NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 13

Canonically ≤n ≤n−1 n Diﬀ (X)/ Diﬀ (X) ' SymC∞(X)(Vect(X)).

3.1.3. In what follows,for x ∈ X, it will be convenient to consider the vector space

Distrx(X) := C ⊗ Diﬀ(X), C∞(X)

≤n n which is the union of its subspaces Distrx (X) := C ⊗ Diff (X). C∞(X) Note that for f ∈ C∞(X) and d ∈ Diff(X), the value of d(f) at x only depends on the image of d in C ⊗ Diff(X). C∞(X)

n ∞ Let Germx be the quotient ring of C (X) by the ideal of functions that vanish to the order 0 n at x (so that evaluation at x deﬁnes an isomorphism Germx ' C). The map d, f 7→ (d(f))(x) deﬁnes a pairing

≤n n (3.1) Distrx (X) ⊗ Germx → C. ≤n It is easy to see that (3.1) is a perfect pairing, i.e., identiﬁes Distrx (X) as the dual of n Germx. Note that for a smooth map F : X → Y,F (x) = y, we have a canonically deﬁned maps

n n (3.2) Diffx(X) → Diffy (Y ), Distrx(X) → Diffy(Y ), ∗ n n where the former map is the dual of the pullback map F : Germy → Germx. Consider the vector space Cn(X) equipped with its natural topology. Consider its topological dual Cn(X)∗. The Taylor expansion at x defines a map

n n C (X) → Germx . Hence, we obtain a naturally deﬁned map

≤n n ∗ (3.3) Distrx (X) ,→ C (X) .

By deﬁnition, C∞(X) := ∩ Cn(X), and it is equipped with the inverse image topology. Set n ∞ ∗ Distrc(X) := C (X) . From (3.3), we obtain an embedding

(3.4) Distrx(X) ,→ Distrc(X).

3.2. Diﬀerentiable vectors. 14 DENNIS GAITSGORY

3.2.1. Let G be a Lie group, and let V be a continuous representation. We shall say that a vector v ∈ V is diﬀerentiable if the function

v v (3.5) F : G → V,,F (g) := Tg(v) is diﬀerentiable at 1.

Lemma 3.2.2. If v ∈ V is differentiable, then the function (3.5) is continuously differentiable. Proof. For ξ ∈ g, let ξr denote the corresponding left-invariant vector field. Note that such fields span the tangent space at every point of G. Hence, it is enough to show that the functions Lieξr (F ) are continuous on G. However, it is easy to see that

v v0 0 v Lieξr (F ) = F , v := dF1 (ξ).

Let V 1 ⊂ V be the subspace consisting of differentiable vectors. By construction, we have a well defined map 1 v g ⊗ V → V, ξ, v 7→ dF1 (ξ). 1 For ξ ∈ g denote the corresponding map V → V by Tξ. Define V n inductively by

n n−1 n−1 V = {v ∈ V ,Tξ(v) ∈ V ∀ξ ∈ g} Set V ∞ := ∩ V n. n We topologize V n inductively as follows as follows. If {ρ} is a system of semi-norms for V n−1 and ξi is a basis for g, we introduce semi-norms ρi on V n by

i ρ (v) = ρ(Tξi (v)).

We give V ∞ the inverse limit topology. In the same way as in Proposition 2.1.6, one shows: Proposition 3.2.3. For any m, n there exists a canonically deﬁned continuous map

n ∗ m+n m C (G) × V → V , µ 7→ Tµ. It is uniquely determined by the requirement that for v ∈ V m+n and v∗ ∈ (V m)∗ we have

∗ ∗ v hv ,Tµ(v)i = hµ, hv ,F ii, where the function hv∗,F vi on G is easily seen to be in Cn(G). As a corollary, we have: Corollary 3.2.4. There is a canonically deﬁned continuous map

∞ ∞ Distrc(G) × V → V , µ 7→ Tµ, compatible with the convolution algebra structure on Distrc(G). NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 15

≤n 3.2.5. Recall the vector space Distr1(G) = ∪ Distr (G). The group law on G endows Distr1(G) n 1 with a structure of associative algebra via (3.2).

We can alternatively view this structure as follows: identify Distr1(G) with left (or right) invariant differential operators on G. Then the associative algebra structure on Diff(G) induces one on Distr1(G). The assignment (ξ ∈ g) 7→ (f 7→ df1(ξ)) defines a map ≤1 g → Distr1 (G) ⊂ Distr1(G), and it is easy to see that it respects the commutator relation. Hence, we obtain a homomorphisms of associative algebras

(3.6) U(g) → Distr1(G). It is easy to see that (3.6) is an isomorphism. Alternatively, we can view (3.6) as an identi- ﬁcation between U(g) with left (or right) invariant diﬀerential operators on G. Combining with (3.2.3) we obtain canonical maps ≤n m+n m U(g) ⊗ V → V , u 7→ Tu as well as an action of U(g) on V ∞.

By construction, for ξ ∈ g ⊂ U(g), this notation is consistent with the notation Tξ introduced earlier. 3.2.6. Here comes the arch-important smoothing construction:

k ∗ Lemma 3.2.7. Let f be an element of Cc (G), and consider f ◦ µHaar as an element of C(G) , where µHaar is a left-invariant Haar measure. Then for any v ∈ V , the vector Tf·µHaar (v) belongs to V k. Proof. It is easy to see that for ξ ∈ g,

Tξ(Tf·µHaar (v)) = TLieξ(f)·µHaar (v). ∞ ∞ Corollary 3.2.8. For f ∈ Cc (G) and any v ∈ V , the vector Tf·µHaar (v) ∈ V belongs to V . Corollary 3.2.9. For any V , the subspace V ∞ ⊂ V is dense.

Proof. We can choose a Dirac sequence of smooth functions fn. By (2.1), we have

Tfn (v) → v.

However, by Corollary 3.2.8, each Tfn (v) is smooth. Corollary 3.2.10. Let V be a continuous ﬁnite-dimensional representation. Then V ∞ = V . Equivalently, the image of MCV : End(V ) → C(G) lies in C∞(G).

Proof. A dense subspace of a ﬁnite-dimensional vector space is everything. 3.3. Compact real algebraic groups. 16 DENNIS GAITSGORY

3.3.1. Let G be an algebraic group over R. We will say that G is compact, if G(R), endowed with its natural structure of Lie group, is compact. Thus, we obtain a functor (3.7) Compact algebraic groups over R → Compact Lie groups. The functor (3.7) is not an equivalence of categories. Indeed, non-isomorphic compact algebraic groups over R can give rise to the same Lie group. Example: Take the group of 3rd roots of unity x3 = 1. The group of its real points is the same as that of the trivial algebraic group.

3.3.2. We shall say that a compact algebraic groups over R is relevant if the map π0(G(R)) → π0(G) is surjective, where π0 is the algebro-geometric group of connected components. We will prove: Theorem 3.3.3. The restriction of the functor (3.7) to the subcategory of relevant compact algebraic groups is an equivalence from the category of relevant compact algebraic groups over R to that of compact Lie groups. 3.3.4. As a ﬁrst step, we will prove:

Theorem 3.3.5. Every compact Lie group admits an injective homomorphism into GLn(R) for some n. Proof. Let K be a compact Lie group. The statement of the theorem is equivalent to fact that K admits a faithful ﬁnite-dimensional representation. Note that if K is a compact Lie group and K ⊃ K1 ⊃ K2... is sequence of closed subgroups with trivial intersection, then then exists an n such that Kn = {1}. The set of isomorphism classes of irreducible representations of K is countable. Enumerate them by ρ1, ρ2, ..., etc. Set Vi := ρ1 ⊕ ... ⊕ ρi and Ki = ker(K → GL(Vi)).

We claim that ∩ Ki = {1}. Indeed, if k ∈ K belongs to the above intersection, then its acts i trivially in every irreducible representation, and hence, by Corollary 2.5.2, multiplication by k acts as identity on C(K). The latter means that k = 1.

Hence, by the above Kn = {1} for some n. 3.3.6. To prove Theorem 3.3.3 we will need the following construction. Let Z be an aﬃne algebraic variety over R, and let X be a subset of Z(R). Let IX be the ideal of regular 0 functions on Z that vanish on X, Let Z := V (IX ) be the corresponding algebraic subvariety in Z. Clearly, X ⊂ Z0(R). 0 0 Moreover, by construction for every connected component of Z0 ⊂ Z , we have 0 (3.8) X ∩ Z0(R) 6= ∅. 3.3.7. Assume now that Z is acted on by a compact Lie group K (i.e., the algebra of regular functions on Z is an algebraic1 representation of K). Assume that X is a single K-orbit.

1algebraic=K-ﬁnite NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 17

Proposition 3.3.8. Under the above circumstances, the inclusion X ⊂ Z0(R) is an isomorphism.

Proof. Let x0 be a point in Z(R) that is not in X. We claim that we can find a regular function that vanishes on X and doesn’t vanish at x0. Let X0 be the K-orbit of x0. We claim that we can find a K-invariant regular function that takes value 0 on X and value 1 on K0. Indeed, by Stone-Weierstrass, the map 0 C[Z] → C(X t X ) is K-equivariant and has a dense image (indeed, embed Z in the affine space Rn so that X t X0 becomes a compact subset of Rn). Averaging with respect to K, we obtain that the map K 0 K C[Z] → C(X t X ) also has a dense image. However, C(X t X0)K ' C ⊕ C, so the latter map is surjective, and in particular, its image contains the element (0, 1). 3.3.9. We are now ready to prove Theorem 3.3.3:

Let now G be a real algebraic group and K a compact subgroup of G(R). Consider the subscheme G0 ⊂ G, obtained by the construction in Sect. 3.3.6. By the functoriality of this construction, we obtain that G0 is an algebraic subgroup of G. By Proposition 3.3.8, the map K → G0(R) is an isomorphism, so that G0 is a compact real algebraic group, and K is the group of its R-points. Moreover, by (3.8), the compact real group G0 is relevant. Combined with Theorem 3.3.5, this implies that that functor in Theorem 3.3.3 is essentially surjective.

Let us now show that the functor in Theorem 3.3.3 is fully faithful. Let G1 and G2 be relevant compact real groups, and let φ : G1(R) → G2(R) be a homomorphism. We wish to show that φ comes from a (unique) homomorphism of algebraic groups. Let

K ⊂ G1(R) × G2(R) 0 be the graph of φ. Let G1 be the subgroup of G1 × G2 corresponding to K via the construction 0 of Sect. 3.3.6. It suﬃces to show that the projection ψ : G1 → G1 is an isomorphism. 0 By Proposition 3.3.8, the map G1(R) → G1(R) is an isomorphism. Hence, ψ induces an isomorphism at the level of Lie algebras, and hence also of the (algebraic) connected components of the identity. Since both groups are relevant, this implies that ψ is itself an isomorphism. Note that the above proof actually showed:

Corollary 3.3.10. Let G1 and G2 be real algebraic groups with G1 compact and relevant. Then the map HomAlgGrp(G1,G2) → HomLieGrp(G1(R),G2(R)) is an isomorphism.

In particular, taking G2 to be GLn,C|R (here the operation Z 7→ Z|R is the operation of restriction of scalars `ala Weil, i.e., the standard way of viewing a complex algebraic variety as a real one), we obtain:

Corollary 3.3.11. For a compact relevant real algebraic group G and its complexiﬁcation GC, the map

HomAlgGrp /C(GC, GLn,C) → HomLieGrp(G(R), GLn(C)) is a bijection. 18 DENNIS GAITSGORY

The last corollary says that for a compact relevant real algebraic group G, the category of algebraic representations of its complexiﬁcation is canonically equivalent to the category of complex representations of the underlying compact Lie group.

4. Week 2, Day 2 (Thurs, Feb 2) 4.1. Compact Lie groups vs complex reductive algebraic groups. 4.1.1. Recall that an algebraic group is said to be reductive if its unipotent radical is trivial.

Proposition 4.1.2. Let G be a compact real algebraic group. Then its compexification GC is reductive. Proof. Let U(G) be the unipotent radical of G. Consider the center of U(G). This is a group n over R, whose complexification is isomorphic to Ga for some n ≥ 1; by Hilbert 90, we obtain that this isomorphism is valid also over R. However, the group of its R-points is then isomorphic to Rn, which cannot be contained as a closed subgroup in a compact group. 4.1.3. From the above proposition we obtain that complexification defines a functor {Compact real algebraic groups} → {Complex reductive groups}. We will study how close this functor is to be an equivalence. 4.2. The polar decomposition. 4.2.1. We will prove the following theorem:

Theorem 4.2.2. Let G a complex algebraic group. Let K be a compact Lie subgroup of G(C). Assume that: (i) The map

kC := C ⊗ k → g R is an isomorphism (i.e., g is a complexiﬁcation of k). (ii) K intersects non-trivially every connected component of G(C); Then the group G carries a unique real structure σ for which K = G(C)σ. Moreover: let p ⊂ g be the subspace {ξ ∈ g, , σ(ξ) = −ξ}. Then the map (4.1) K × p → G(C), k, p 7→ k · exp(p) is a diﬀeomorphism.

In the situation of the theorem we denote by P ⊂ G(C) the image of p ⊂ g under the exponential map. By (4.1), this is a closed submanifold of G(C). By construction −1 (4.2) P ⊂ Pe := {g ∈ G(C), , σ(g) = g }. Corollary 4.2.3. (a) The product map deﬁnes a diﬀeomorphism t ({k} × pk) → P.e k∈K,k2=1

(b) For every g ∈ Pe, we have g2 ∈ P . NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 19

Corollary 4.2.4. Let G be as in Theorem 4.2.2. Then G is reductive.

Proof. Follows from Proposition 4.1.2.

Corollary 4.2.5. The map K → G(C) is a homotopy equivalence. In particular π0(K) ' π0(G(C)), and π1(K) ' π1(G(C)).

Note that for an algebraic variety Z over C, the topological π0(Z(C)) identifies with the algebro-geometric π0(Z). Also, for a reductive algebraic group G one can define its algebraic π1(G), and for G over the field of complex numbers we have π1(G(C)) ' π1(G). Corollary 4.2.6. In the situation of Theorem 4.2.2, we have

(4.3) ZG(C) = CentrG(C)(K) = ZK × (ZG(C) ∩ P ). If G is semi-simple, ZG(C) = ZK . Proof. By the isomorphism (4.1), in order to prove (4.3), we only need to show that the inclusion

ZG(C) ⊂ CentrG(C)(K) is an equality. If g ∈ Centr(K), then its adjoint action on k is trivial, and hence its adjoint action on g is also trivial (by condition (i)). Therefore g ∈ CentrG(C)(G0(C)). To prove that the adjoint action of g on all of G is trivial, it is therefore enough to show that every connected component of G(C) contains a point that commutes with g. However, every connected component contains a point of K by (ii).

Let us show that if G is semi-simple, then ZG(C) ∩ P is trivial. Indeed, by the above, ZG(C) ∩ P equals the set of AdK -invariants in P , and the exponential map identiﬁes the latter with pK . However, pK ⊂ gK = gG, and the latter is trivial, since G is semi-simple. Corollary 4.2.7. In the situation of Theorem 4.2.2, we have

NormG(C)(K) = K × (ZG(C) ∩ P ).

Proof. It is clear that NormG(C)(K) = K · (Norm(K) ∩ P ). However, from (4.1) it follows that Norm(K) ∩ P ⊂ CentrG(C)(K), and the latter equals ZG(C). n n × n 4.2.8. Example. Let G = Gm. Consider the subgroup K = U1 ⊂ (C ) . It obviously satisﬁes the assumption of Theorem 4.2.2. The corresponding real structure is σ(g) = g−1. n We claim that this is the unique compact real form of Gm. × n n n Indeed, the quotient (C ) /U1 is isomorphic to R , and hence contains no non-trivial compact subgroups. Hence, if σ0 is another compact real structure, then the corresponding subgroup 0 0 0 0 K is contained in K. However, since dimR(k ) = n, we obtain that K = K. Hence, σ = σ. 4.3. Proof of Theorem 4.2.2.

4.3.1. We ﬁrst consider the particular case when G = GL(V ), where V is a complex vector space. Assume that V is equipped with a Hermitian form. Deﬁne a real structure on G = GL(V ) by σ(S) = (S†)−1.

Then K = U(V ) and p = E(V ) is the space of Hermitian operators. The exponential map defined an isomorphism E(V ) → E+(V ), 20 DENNIS GAITSGORY where the latter is the space of positive Hermitian operators. The statement that + U(V ) × E (V ) → GL(V, C) is an isomorphism is well-known. The inverse map is constructed as follows: For S ∈ GL(V, C) consider S ◦ S†. This is a positive Hermitian operator; hence it admits a † 1 well-defined square root (S ◦ S ) 2 . In sought-for inverse map sends S to the pair † − 1 † 1 + (S ◦ S ) 2 ◦ S ∈ U(V ), (S ◦ S ) 2 ∈ E (V ). 4.3.2. Let us now return to the general case of Theorem 4.2.2. First off, it is clear that the real structure σ is unique: indeed, condition (i) fixes what it is on g, and hence on G0, and condition (ii) fixes it on the rest of G. Let G be as in the theorem, and let G → GL(V ) be a faithful representation. Choose a K- invariant Hermitian structure on V . We claim that the real structure on GL(V ) from Sect. 4.3.1 induces one on G. We have to show that the operation S 7→ (S†)−1 preserves the image of G(C) in GL(V, C). By construction, it acts as identity on K, and hence preserves k, and hence by (i) all of g, and hence G0(C). By (ii) it preserves all of G(C). Denote Ke = G(C)σ. We obviously have K ⊂ Ke, and this inclusion is an isomorphism at the level of Lie algebras, and hence on the neutral connected components (we will soon prove that it is an isomorphism also at the group level). 4.3.3. We claim that the mutually inverse maps + U(V ) × E (V ) GL(V ) send the subsets G(C) ⊂ GL(V ) and Ke × p ⊂ U(V ) × E+(V ) to one another. To show this, we only need to see that for g ∈ G(C) ∩ E+(V ), the elements gt ∈ GL(V, C), t ∈ R and log(g) ∈ End(V ) belong to G(C) and g, respectively. If n ∈ Z, then gn clearly belongs to G(C). However, for any S ∈ GL(V, C), the Zariski closure of the elements Sn contains the elements St, t ∈ R and log(S). This implies that gt and log(g) belong to G(C), since G(C) is Zariski-closed in GL(V, C). 4.3.4. Thus, we have established that the map Ke × p → G(C). is a diffeomorphism.

In particular π0(Ke) → π0(G(C)) is an isomorphism. However, by assumption, the composite map π0(K) → π0(Ke) → π0(G(C)) is surjective. Hence, the map K → Ke is a surjective at the level of π0 components, and hence is an isomorphism. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 21

4.3.5. Example. Let V be a real vector space equipped with a positive-deﬁnite scalar product. Consider the corresponding algebraic group O(V ). Then this is a relevant compact real form of its complexiﬁcation O(V )C. Indeed, we only need to see that O(V, R) hits every connected component of O(V, C), but the latter is obvious.

5. Week 3, Day 1 (Tue, Feb. 7) 5.1. Compact Lie groups vs compact Lie algebras.

5.1.1. Recall that a Lie algebra g is semi-simple if and only if the Killing form

Kil(ξ, η) := Tr(adξ ◦ adη, g) is non-degenerate. (This is insensitive to what ﬁeld we are working over, as long as it is of characteristic 0.)

5.1.2. Let k be a real Lie algebra. We shall say that it is compact if its Killing form is negative deﬁnite. By the above, if k is compact, then it is semi-simple. Let Aut(k) be the real algebraic group of automorphisms of k. We claim: Lemma 5.1.3. (a) If k is compact, then the Lie group Aut(k)(R) is compact. (b) If K is a compact Lie group with trivial center, then its Lie algebra is compact.

Proof. By definition, Aut(k)(R) is a closed subgroup of the group GLn(R) of linear automorphisms of k as a vector space, where n = dim(k). However, it is actually contained in the orthogonal group On(R) corresponding to Kil. Since the latter is negative-definite, the orthogonal group in question is compact. Hence, Aut(k)(R) is compact as well, proving (a). For (b), since K is compact, we can choose a K-invariant positive-definite scalar product on k. We have an embedding

K → On,

Consider the corresponding maps of Lie algebras k ,→ on.

The Killing form on k is obtained by restriction form the standard form on on:

(S1,S2) 7→ Tr(S1 ◦ S2), while the above form is easily see to be negative-deﬁnite (skew-symmetric matrices have imaginary eigenvalues). Corollary 5.1.4. For a compact Lie algebra k, the neutral (algebraic) connected component Aut(k)0 of Aut(k) is a relevant compact real algebraic group. 5.1.5. Let g be a complex semi-simple Lie algebra, with a chosen Cartan and Boreal subalgebras h ⊂ b ⊂ g. For vertex root α let hα ∈ h be corresponding simple coroot element. Pick a non-zero element eα in the corresponding root space. Let fα be the unique element in the corresponding negative root space such that [eα, fα] = hα. Then it is easy to see that there exists a unique real structure σ on g such that

σ(hα) = −hα; σ(eα) = −fα; σ(fα) = −eα. 22 DENNIS GAITSGORY

We claim that this real structure is compact. Indeed, it is well-known that the Killing form is positive definite on Span (h ) and R α Kil(h , h ) Kil(e , f ) = α α > 0. α α 2 Hence, it is negative definite on i · h as well as on eα − fα and i · (eα + fα), while the rest of the scalar products are zero. 5.2. Existence and uniqueness of the compact real form. 5.2.1. Let σ and τ be two real structures on a complex Lie algebra. We shall that they are compatible if the (sesqui)-linear automorphisms σ and τ commute. This is equivalent to requiring: (i) gσ = (gσ ∩ gτ ) ⊕ (gσ ∩ i · gτ ); (i’) gτ = (gσ ∩ gτ ) ⊕ (i · gσ ∩ gτ ); (ii) σ(gτ ) = gτ ; (ii’) τ(gσ) = gσ. (iii) The linear automorphism θ = τ ◦ σ satisfies θ2 = 1. Note that if both σ and τ are compact, then σ = τ (indeed, if the Killing form is negative- definite on gσ, then it is positive-definite on i · gσ, so the intersection i · gσ ∩ gτ is zero). 5.2.2. We have: Theorem 5.2.3. Let g be a complex semi-simple Lie algebra, and let σ be a compact real structure on g. Let τ be some other real structure on g. (a) There exists an element g ∈ Aut(g)0(C) such that Adg(σ) is compatible with τ. (b) If σ1 and σ2 be two compact real structures on g compatible with a real form ψ, then there ψ exists g ∈ Aut(g )(R)0 such that Adg(σ1) = σ2.

Corollary 5.2.4. If σ1 and σ2 are two compact real structures on a semi-simple Lie algebra g are conjugate by an element of Aut(g)0(C). Proof. Set G = Aut(g). Let K ⊂ G(C) be the group of fixed points of the involution induced by σ. By Lemma 5.1.3, K is compact. We’d like to apply Theorem 4.2.2. For this we need to know that K meets every connected component of Aut(g). This is obvious for the compact real form from Sect. 5.1.5. So we argue as follows: we first prove the theorem in this case (below); then deduce Corollary 5.2.4, thereby verifying the condition on π0 for any given compact form. Then the proof given below applies to this compact form. Consider the decomposition G(C) ' K × P. Recall also the subset Pe ⊂ G(C), see (4.2) The element θ = τ ◦ σ ∈ G(C). We have g ∈ Pe. Hence, by Corollary 4.2.3, θ2 ∈ P . Set 1 0 g = θ 4 ∈ P . Then it is easy to see that σ = Adg(σ) satisfies τ ◦ σ0 = σ0 ◦ τ. This proves point (a).

For point (b), we apply the above construction for σ = σ1 and τ = σ2. Then all the elements gt (for t ∈ R) commutes with ψ, and hence belong to (the identity component of) Aut(gτ )(R). NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 23

5.2.5. We will now prove:

Theorem 5.2.6. Let G be a complex reductive group.

(a) G admits a relevant compact real form, and such forms are in bijection with those of G0/ZG0 .

(b) Any two such real forms are conjugate by an element of G0(C).

Proof. Note that if G is adjoint (i.e., maps isomorphically to Aut(g)0), then the assertion of Theorem 5.2.6 follows from Sect. 5.1.5 and Corollary 5.2.4.

Assume that G is such that G0 is adjoint. Choose a compact real form of G0; let K0 denote the corresponding compact Lie group. Set K = NormG(C)(K0). Then K · G0(C) = G(C) and K ∩ G0(C) = K0, by Corollaries 5.2.4 and 4.2.7. Applying Theorem 4.2.2, we obtain that the result follows from the adjoint case.

Let G be a torus. Then G admits a unique compact real form by Sect. 4.2.8.

Let G be such that G0 is a torus. In this case it is still true that G has a unique compact real form. Indeed, the quotient of G(C) by the maximal compact subgroup of G0(C) is canonically of the form n π0(G) n R .

Let G be an arbitrary reductive group. Consider the (surjective) map

0 G → Ge := G/ZG0 × G/(G0) . π0(G)

Choose a relevant compact real structure on G/ZG0 and consider the unique compact real 0 structure on G/(G0) . Let Ke be the corresponding compact Lie subgroup of Ge(C). Let K be the preimage of Ke in G(C). Applying Theorem 4.2.2, we obtain the result. (Indeed, any real 0 structure on G induces one on G/ZG0 and G/(G0) by the canonicity of these quotients.)

5.3. Polar decomposition of a real reductive group.

5.3.1. Let G be a complex reductive group, equipped with a real form τ. We will write GR for the corresponding algebraic group over R and G(R) for its set of real points, i.e., τ G(R) = G(C) .

We shall say that a relevant compact real form σ is compatible with τ if θ = τ ◦ σ = σ ◦ τ as automorphisms of G. Note that in this case θ deﬁnes an involution of GR as a real algebraic group.

From Theorem 5.2.6 and Theorem 5.2.3 we obtain:

Corollary 5.3.2. Let G be as above. Then G admits a relevant compact real form compatible with τ; such forms are in bijection with those on G0/ZG0 . Any two such real forms are conjugate by an element of G(R)0. 24 DENNIS GAITSGORY

τ 5.3.3. Let gR be the Lie algebra of GR, i.e., gR := g . For a compatible compact real form σ, consider the subspaces kτ = gθ = (g ∩ k ) = gσ = kθ R R C R and τ p = {ξ ∈ gR , θ(ξ) = −ξ} = (gR ∩ pC) = {ξ ∈ gR , σ(ξ) = −ξ} = {ξ ∈ pC , θ(ξ) = −ξ}, where g ' kC ⊕ pC is the decomposition corresponding to σ. We have: τ τ gR ' k ⊕ p . If g is semi-simple, then the form − Kil(ξ, θ(η)) is positive-deﬁnite on gR. 5.3.4. Let K = G(C)σ. Consider the subgroup τ θ θ σ K = K = G(R) ∩ K = G(R) = G(R) .

From Theorem 4.2.2 we obtain: Theorem 5.3.5. The map (k, p) 7→ k · exp(p) deﬁnes a diﬀeomorphism τ τ K × p → G(R).

5.3.6. Example. Let V be a real vector space, and GR = GL(V ). Choose a positive-deﬁnite scalar product on V , and endow VC with the corresponding Hermitian structure.

The real form τ on G := GL(VC) is given by S 7→ S. The compatible compact form σ is given by S 7→ (S†)−1. We have: K = U(V ) and Kτ = O(V ).

5.3.7. As in the complex case, we obtain:

Corollary 5.3.8. The inclusion Kτ → G(R) is a homotopy equivalence. In particular τ τ π0(K ) ' π0(G(R)), and π1(K ) ' π1(G(R)). τ τ Corollary 5.3.9. NormG(R)(K ) = K × (ZG(R) ∩ PR). τ Kτ Proof. We only need to show that (p ) is contained in the center of gR. Thus, we can assume τ Kτ 0 that gR is semi-simple. Let ξ be an element of (p ) . We need to show that [ξ, ξ ] = 0 for any 0 τ 0 τ 0 τ ξ ∈ p . However, [ξ, ξ ] ∈ k , and it is suﬃcient to show that Kilg([ξ, ξ ], η) = 0 for any η ∈ k . However, 0 0 Kilg([ξ, ξ ], η) = Kilg(ξ , [ξ, η]), while [ξ, η] = 0 by assumption.

6. Week 3, Day 2 (Thurs, Feb. 9) 6.1. The maximal compact subgroup. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 25

6.1.1. Let GR be a real reductive group, and let σ be a compatible relevant compact real form of GC. Consider the corresponding subgroup τ σ K = K ∩ G(R) = G(R) and the subset τ P = P ∩ G(R). We will prove: Theorem 6.1.2. For any compact subgroup K0 ⊂ G(R), there exists an element g ∈ P τ such 0 τ that Adg(K ) ⊂ K . We will deduce this theorem from the next one.

6.1.3. Note that G(R) acts on P τ by the formula g(p) = g · p · σ(g−1). The stabilizer of 1 ∈ P τ is by definition G(R)σ = Kτ . We will prove: Theorem 6.1.4. Any compact subgroup K0 ⊂ G(R) has a fixed point on P τ . Let us see how Theorem 6.1.4 implies Theorem 6.1.2: Proof of Theorem 6.1.2. Let p ∈ P τ be the fixed point of K0. Conjugating K0 by p we can assume that K0 stabilizes 1 ∈ P τ . But this means that K0 ⊂ Kτ .

6.2. Proof of Theorem 6.1.4. To prove the theorem, we can quotient G out by ZG0 , so we can assume that G is semi-simple. We will define a certain continuous function τ τ >0 r : P × P → R with the following properties: (i) r is G(R)-invariant with respect to the diagonal action of G(R) on P τ × P τ ; (ii) For every fixed p0 ∈ P τ and sufficiently large R ∈ R>0 (depending on p0), the function 0 τ >0 p 7→ r(p, p ): P → R takes values > R away from a compact subset. (iii) For every fixed p0, p ∈ P τ , the function t 0 >0 r(p , p ): R → R is strictly convex. 6.2.1. Let us prove Theorem 6.1.4 assuming the existence of such a function. Proof of Theorem 6.1.4. For every compact subset Ω ⊂ P τ , define the function τ >0 0 ρΩ : P → R , ρΩ(p) = max r(p, p ). p0∈Ω

Lemma 6.2.2. The function ρΩ has a unique point of minimum. Let us prove the theorem assuming this lemma. Indeed, take Ω to be the orbit of 1 ∈ P τ under K0. Let p ∈ P τ its (unique) point of minimum. Since Ω is K0-invariant, by condition (i), 0 0 the function ρΩ is also K -invariant. Hence, p is K -invariant. 26 DENNIS GAITSGORY

6.2.3. Proof of Lemma 6.2.2. The function ρΩ is easily seen to be continuous. By condition >0 (ii), for all suﬃciently large R ∈ R such that the function ρΩ takes values > R outside a compact set. Hence, it attains a minimum.

Assume that it has two minima, p1 and p2. Translating by means of p1, we can assume that p1 = 1 and p2 = p. Consider the function

t ρΩ(p ), R → R. By assumption, it has a minimum at t = 0 and t = 1. However, it follows from condition (iii) that the above function is strictly convex, which is a contradiction.

6.2.4. We will now construct the desired function r; we will obtain it by restriction on G(C). Choose a G → GL(n), and choose a K-invariant Hermitian structure on Cn, so that we are in the situation of Sect. 4.3.2. Thus, we can assume that G = GL(n) and P ' E+(n).

Since G was semi-simple, its image in GL(n) in fact belongs to SL(n).

We set −1 r(S1,S2) = Tr(S1 ◦ S2 ).

Property (i) is evident. For property (ii), we claim that for a ﬁxed S2, we have

r(S1,S2) > b · ||S1|| for some constant b > 0. Indeed, choose an orthonormal basis in which S1 is diagonal. Then

r(S1,S2) = Σ ai,i · bi,i, i

−1 where ai,i are the diagonal entries of S1 and bi,i are the diagonal entries of S2 . Note that all bi,i are strictly positive, and

||S1|| = max ai,i.

This implies the claim: take b = min bi,i and take the minimum over all possible orthonormal basis (the set of such is compact).

Now, we note that on the set SL(n, C), the sets {S, ||S|| ≤ c} are compact (indeed, SL(n, C) is a closed subset of Mat(n × n, C). Property (iii) follows similarly:

t t r(S1,S2) = Σ ai,i · bi,i. i

7. Week 4, Day 1 (Tue, Feb. 14) 7.1. Elements in a reductive group/algebra. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 27

7.1.1. Let G be a connected reductive group. We will prove: Theorem 7.1.2. Every element in a reductive group G (over an algebraically closed ﬁeld) can be conjugated into a given Borel. Proof. Let X be the ﬂag variety (i.e., X = G/B), thought of as the variety of Borel subgroups. Over it we consider the bundle, denoted Ge (called the Grothendieck alteration) that attaches to a given x ∈ X the corresponding subgroup Bx. We have the natural forgetful map π : Ge → G. Since G acts on X transitively, our assertion is equivalent to the fact that π is surjective. The latter will follow from the combination of the following three observations: (i) G is reduced and irreducible (obvious); (ii) The image of π is closed. This follows from the fact that π is proper; in fact, it’s the composition of the closed embedding G,e → X × G and the projection X × G → G. (iii) π is generically smooth. To see the latter, take the point of Xe of the form (x, ξ), where ξ is regular semi-simple. Consider the map

bx ⊕ g → T(x,ξ)(X), where the first component is the tangent space to the fiber of Ge → X and the second component is given by the action of G on Ge. We claim that the composition dπ bx ⊕ g → T(x,ξ)(X) → Tξ(G) ' g is surjective. Indeed, this composition is given by the inclusion of bx into g (the first component), and the map ξ 7→ Adx(ξ) − ξ

(the second component). The assertion follows now from the fact that the action of Adx − Id on g/bx is surjective: indeed the regularity assumption on x means that it has no eigenvalue 1 on Adx. 7.1.3. The same proof shows that every element in g can be conjugated into a given Borel subalgebra.

Recall that an element of G is said to be semi-simple (resp., nilpotent) if its action on OG (by left translations) is semi-simple (resp., nilpotent). Equivalently, if its action in every ﬁnite- dimensional representation is semi-simple (resp., nilpotent). The latter point of view implies that the condition of being semi-simple (resp., nilpotent) survives under homomorphisms of groups. Corollary 7.1.4. Every semi-simple (resp., nilpotent) element of G can be conjugated into T (resp., N). Proof. By Theorem 7.1.2, we can conjugate our element into B. Now the claim is that every semi-simple (resp., nilpotent) element of a solvable group can be conjugated into a given maximal torus (resp., is contained in the unipotent radical). Corollary 7.1.5. Every torus on G can be conjugated into T . Proof. Every torus is generated (i.e., is the Zariski closure of the abstract group generated) by a semi-simple element. 28 DENNIS GAITSGORY

7.1.6. Let K be a maximal compact in G(C). Recall that

TK := K ∩ B(C) is a maximal compact subgroup of T (C) for some Cartan T ⊂ B.

Theorem 7.1.7. Every element of K can be conjugated into TK . Proof 1. Suppose by contradiction that k ∈ K is an element that cannot be conjugated into TK . Consider the action of k on K/TK ' G(C)/B(C) = X(C). We obtain that this action has no fixed points. Now, the Lefschetz fixed point theorem implies that the action of k on H•(X(C), R) has a zero trace. Now, it is easy to see that since K is connected, the traces of all elements of K on H•(X(C), Z) are equal (if two endomorphisms of a space are homotopic, they induce the same maps on cohomology). Hence, they are all equal to zero. Let us take k0 = 1. Then the trace in question equals χ(X(C)) = |W |= 6 0, which is a contradiction. 0 Alternatively, we can take k to be a regular element in TK . The automorphism of X(C) defined by such a k0 preserves the orientation and has isolated fixed points (the W Borels that contain T ). Hence, the Lefschetz number equals |W |. Proof 2, due to Sherry Gong. We will emulate the proof of Theorem 7.1.2. Set Ke be the fibration over X(C) that attaches to every x the intersection Bx ∩ K. We have the evident map π : Ke → K.

Since the action of K on X(C) is transitive, it is enough to show that π is surjective. Note that this is a map between orientable compact manifolds. In the proof of Theorem 7.1.2 we saw that π is an orientation-preserving local isomorphism on a neighborhood of a regular element, and the preimage of such an element has has |W | many preimages. We now have the following general assertion (well-deﬁnedness of degree): Lemma 7.1.8. Let us be given a map f : Y → Z between orientable compact manifolds. Suppose that Z contains a a point z such that f is an orientation-preserving local isomorphism on a neighborhood of z. Then f is surjective.

7.2. Real tori. Let T be a real torus; let GC denote its complexiﬁcation. We denote by t the Lie algebra of T and by tC its complexiﬁcation, i.e., the Lie algebra of TC. 7.2.1. We have:

TC ' Gm ⊗ Λ Z for a canonically deﬁned lattice Λ. The real structure τ on TC corresponds to an involution θ on Λ: τ(c ⊗ λ) = c ⊗ θ(λ), so that τ ◦ θ is the unique compact real structure on TC. × >0,× Write Gm(C) = C as U(1) × R . Hence, τ >0,× τ T (R) = (U(1) ⊗ Λ) × (R ⊗ Λ) . Z Z NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 29

Note that τ acts on U(1) ⊗ Λ by Z τ(c ⊗ λ) = c−1 ⊗ θ(λ) and on (R>0,× ⊗ Λ)τ by Z τ(c ⊗ λ) = c ⊗ θ(λ).

The exponential map identiﬁes R>0,× with the additive group R, so that >0,× R ⊗ Λ ' R ⊗ Λ, Z Z where the latter is a vector space, equipped with a linear involution θ. Thus >0,× τ θ (R ⊗ Λ) ' (R ⊗ Λ) , Z Z where the latter is the subspace of θ-invariants. 7.2.2. We denote >0,× τ τ A := (R ⊗ Λ) and a := (R ⊗ Λ) , Z Z so that the exponential map identiﬁes a and A. τ KT := (U(1) ⊗ Λ) , Z i.e., KT ⊂ T (R) is the maximal compact. Thus:

(7.1) T (R) = KT × A. This is the polar decomposition of Theorem 5.3.5 for T (R). 7.2.3. We shall say that an element in T (R) is split if it belongs to the A factor in (7.1). We 0 0 shall say that a subtorus T ⊂ T is split if θ|T 0 is trivial (equivalently, KT ∩ T (R) is ﬁnite). Consider the corresponding decomposition

(7.2) t ' kT ⊕ a.

We shall say that an element of k is split if it belongs to the a factor in (7.2). We shall say that a subspace v ⊂ t is split if all of its elements are split. Note that a subtorus T 0 is split if and only its Lie algebra t0 is split. From the above analysis it is easy to deduce the following: Proposition 7.2.4. (a) An element in T (R) is split if and only if the algebraic group that it generates is connected (and hence is a torus), whose Lie algebra is split. (a’) An element in T (R) is split if and only if its action on every (algebraic) representation has positive real eigenvalues. (b) An element in ξ ∈ t is split if and only if the smallest subtorus T 0 ⊂ T such that ξ ∈ t0, is split. (b’) An element in t is split if and only if its action on every (algebraic) representation has real eigenvalues. 7.3. Split elements in a real reductive group. In this subsection we let G be a real reductive group, let GC denote its complexiﬁcation. We denote by g the Lie algebra of G and by gC its complexiﬁcation, i.e., the Lie algebra of GC. 30 DENNIS GAITSGORY

7.3.1. Let g ∈ G be an element. We shall say that g is split if: (i) it is semi-simple, (ii) the algebraic group that it generates is a split torus.

Let ξ ∈ g be an element. We shall say that ξ is split if it is semi-simple and smallest torus T 0 ⊂ G such that ξ ∈ t0, is split.

We shall say that a subalgebra v ⊂ g is R-diagonalizable if it is abelian and consists of split elements.

From Proposition 7.2.4 we obtain:

Corollary 7.3.2. (a) An element g ∈ G if and only if its action on every (algebraic) representation of G is diagonalizable (over R) with positive eigenvalues. (b) A subalgebra v ⊂ g is R-diagonalizable if and only if its action on every (algebraic) representation of G is simultaneously is simultaneously diagonalizable (over R). (c) For an R-diagonalizable subalgebra v ⊂ g, the smallest subtorus Tv ⊂ G such that v ⊂ Tv, is split.

7.3.3. We now claim:

Proposition 7.3.4. (a) Let v ⊂ g be an R-diagonalizable subalgebra. Then there exists a compatible compact form on GC such that v ⊂ p. (b) If v is contained in p, then v is R-diagonalizable. Proof. We will use the following statement, which can be proved by an argument similar to that of Theorem 5.2.3:

Theorem 7.3.5. Let φ : G1 → G2 be a homomorphism of real reductive groups. Then given 1 2 a compact form σ1 on G compatible with G1, one can find a compact form σ2 on G that is 2 C C compatible with G and σ2 ◦ φ = φ ◦ σ1. For point (a), let G1 be its centralizer in G, and let g0 be its Lie algebra. Then G1 is a reductive group. By Theorem 7.3.5 it suffices to find a compatible compact form for G1.

1 1 0 1 We have g ' zg1 ⊕ (g ) . Choose any compact form on G . We have

1 0 p = (p ∩ zg1 ) ⊕ (p ∩ (g ) ).

Then by Proposition 7.2.4 and (7.2), we have v ⊂ (p ∩ zg1 ) ⊂ p. For point (b), consider a representation G → GL(V ). It suﬃces to show that every element of p acts by an R-diagonalizable operator on V . By Theorem 7.3.5, we can choose a positive deﬁnite scalar product on V such that p maps to E(V ). This implies the claim.

8. Week 4, Day 2 (Thurs, Feb. 16) 8.1. The maximal R-diagonalizable subalgebra. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 31

8.1.1. We shall say that a subalgebra a ⊂ g is a maximal R-diagonalizable subalgebra if it is an R-diagonalizable subalgebra and is not properly contained in other such. Let a be a maximal R-diagonalizable subalgebra. Let Ta be the smallest split subtorus of G such that a ⊂ Lie(Ta). By Corollary 7.3.2 and the maximality assumption the above inclusion is in fact an equality a = Lie(Ta).

This establishes a bijection between maximal R-diagonalizable subalgebras of g and maximal split subtori in G. Set A := Ta(R)0; this is the subgroup of Ta(R) as in (7.1). 8.1.2. We claim:

Proposition 8.1.3. Given a maximal R-diagonalizable subalgebra a ⊂ g and an arbitrary R- diagonalizable subalgebra v ⊂ g, there exists an element of G(R)0 that conjugates v into a. Proof. By Proposition 7.3.4(a), we can assume that both v and a are contained in p. Choosing a generic element ξ ∈ v, its suﬃces to show that there exists k ∈ K such that Adk(ξ) ∈ a; by the maximality of a, the latter is equivalent to the condition that [Adk(ξ), η] = 0 for a given generic element η ∈ a. Consider the function

f(k) = (Adk(ξ), η). By compactness, choose a point k ∈ K, on which it attains a maximum. We claim that this point will do. Indeed, conjugating ξ by k, we can assume that k = 1. We obtain that the function f has a zero diﬀerential at k = 1. However, the diﬀerential is given by k 7→ ([k, ξ], η) = (k, [ξ, η]). We have [ξ, η] ∈ k, and (−, −) is non-degenerate on k. Hence, [ξ, η] = 0. 8.1.4. We shall say that the real group G is split if a is a Cartan subalgebra of g. One can show that every connected complex reductive group has a unique split form (up to conjugacy).

8.2. The centralizer of a maximal R-diagonalizable subalgebra. 8.2.1. As is the case of any abelian semi-semi subalgebra in g, the centralizer of a in g is the Lie algebra m of a Levi subgroup M.

Proposition 8.2.2. m ' a ⊕ (m ∩ k). Proof. Let ξ + η be an element of g that centralizes a, with ξ ∈ k and η ∈ p. It suﬃces to show that ξ and η each centralize a. For a ∈ a we have 0 = [a, ξ + η] = [a, ξ] + [a, η], where [a, ξ] ∈ p and [a, η] ∈ k. Hence, both are zero. Corollary 8.2.3. (a) The map A × (K ∩ M) → M is the polar decomposition of Theorem 5.3.5 for M.

(b) The map A × (K ∩ ZM ) → ZM is the polar decomposition of Theorem 5.3.5 for ZM . Corollary 8.2.4. The group M is compact modulo its center. 32 DENNIS GAITSGORY

8.3. The minimal parabolic. 8.3.1. By assumption, the adjoint action of a on g is diagonalizable. Write M g = m ⊕ gα, α where α ∈ a∗. Note that since σ acts as −1 on a, we have

τ(gα) = g−α. One can show that the collection of those α that appear forms a root system. 8.3.2. In terms of this decomposition, k identiﬁes with the direct sum of m and the span of elements (ξα + ξ−α), ξα ∈ gα, ξ−α ∈ g−α, τ(ξα) = ξ−α. 8.3.3. Choose an element a ∈ a such that α(a) 6= 0 for any α. Then the subspace M q := m ⊕ gα, α,α(a)>0 is the Lie algebra of a parabolic subgroup P of G. (This manipulation is valid for any abelian semi-simple subalgebra in g.) Let n denote the Lie algebra of N(Q). We have:

n = ⊕ gα. α,α(a)>0 Proposition 8.3.4. The subgroup Q is minimal among parabolics deﬁned over R. Proof. The statement is equivalent to the fact that M has no proper parabolics deﬁned over R. This follows from the fact that M/Z(M) is compact (compact real groups do not have proper parabolics because they have no unipotent elements). 8.3.5. Consider the subgroup 0 Q ⊂ P (R) equal to the preimage of A ⊂ Ta(R) ⊂ ZM (R) ⊂ M(R) under the projection Q(R) → M(R). By construction, P 0 is an extension 0 1 → Nmin(R) → Q → A → 1. 8.3.6. We now claim:

Theorem 8.3.7. Any unipotent subgroup of G can be conjugated into Nmin by an element of G(R). Proof. Let N 0 ⊂ G be a unipotent subgroup. By Hilbert 90, it is enough to see that N 0(R) can be conjugated into N(R). Consider the quotient G(R)/Q(R). We claim that is it enough to show that N 0(R) has a fixed point on G(R)/Q(R). Indeed, such a point, will conjugate N 0(R) into Q(R). Now, any unipotent element of Q(R) is contained in Nmin(R). Indeed, the quotient Q(R)/Nmin(R) ' M(R) contains no unipotent elements. To prove the existence of the fixed point, we will show that for a proper variety X and a unipotent group N 0 acting on (everything is defined over R), every connected component of X(R) contains an N 0(R)-fixed point. The proof is obtained as in the case of algebraically closed fields: NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 33

0 Reduce by induction to the case when N = Ga. Then the claim is that for any x ∈ X(R), the map 1 A → X, a 7→ a · x extends to a map P1 → X (by the valuative criterion). Its value on ∞ ∈ P1(R) is the desired ﬁxed point. 8.4. The Iwasawa decomposition.

8.4.1. We will prove: Theorem 8.4.2. The product map 0 Q × K → G(R) is a diﬀeomorphism. 8.4.3. For the proof we ﬁrst notice that the corresponding assertion does take place at the Lie algebra level: Proposition 8.4.4. The sum map a ⊕ n ⊕ k → g is an isomorphism. Proof. Follows from Sect. 8.3.2. We can now prove Theorem 8.4.2:

Proof. We need to show that the map 0 (8.1) K → G(R)/Q , given by the action on the element 1 ∈ G(R)/Q0 is a diﬀeomorphism. Since K is compact, the image is closed. We now claim that the map in question is sub- mersive. This is an assertion at the level of tangent spaces, and follows from the surjectivity of the map in Proposition 8.4.4. Hence, the image is open. Being both open and closed, it is the union of certain connected components. However, the map (8.1) is a bijection at the level of connected components (by Theorem 5.3.5).

Thus, we obtain that the action of K on G(R)/Q0 is transitive. It remains to see that 0 StabK (1) = 1. However, this is just the fact that K ∩ P = 1. Remark 8.4.5. The action of K on G(R)/Q(R) comes from an action of the complex algebraic group K = Gθ on the complex algebraic variety (G/Q) . We note, however, that the latter C C C action is not necessarily transitive. However, the above proof shows that the orbit of the element

1 ∈ (G/Q)C under this action is open.

Here is how it looks in an example G = SL2,R. We have Q = B; K = U(1). At the 1 1 complexiﬁed level GC = SL2 and KC = Gm;(G/P )C ' P . The action of Gm on P is the usual one, and the open orbit in question is P1 − {0, ∞}. 8.5. The Cartan decomposition. 34 DENNIS GAITSGORY

8.5.1. Consider the product map

(8.2) K × A × K → G(R). The Cartan decomposition is the following statement:

Theorem 8.5.2. The map (8.2) is surjective and proper (the preimage of every compact is compact).

The meaning of this theorem is that G(R) ”looks like A modulo something compact”. Remark 8.5.3. The map (8.2) is (obviously) not bijective. For example, the subset

K × {1} × K ⊂ K × A × K → G(R) gets collapsed onto just one copy of K.

8.5.4. Proof of properness. It suﬃces to show that the map K × A → G(R)/K is proper. This follows from the fact that the map A → G(R)/K is proper (by Theorem 8.4.2) and the fact that K is compact.

8.5.5. Proof of surjectivity. By the polar decomposition, the assertion is equivalent to the fact that the adjoint action of K on P deﬁnes a surjection

K × A → P, or, equivalently, K × a → p.

I.e., we need to show that every element of p can be conjugated by means of K into a. But we have shown that in Proposition 8.1.3.

9. Week 5, Day 1 (Tue, Feb. 21) 9.1. Admissible representations. Let G be a real reductive group. We will assume that G is connected as an algebraic group (which does not imply that the corresponding Lie group is connected. To save the notation we will denote by the same symbol G (rather than G(R)) the corresponding Lie group. Let K ⊂ G be its maximal compact. Note that K may be disconnected.

9.1.1. Let V be a continuous representation of G. Recall that V is acted on by the algebra Measc(G) by convolutions.

Recall that we denote by V ∞ ⊂ V the vector subspace consisting of smooth vectors. We have shown that it is dense. We recall also that V ∞ is acted on by a larger algebra, namely ∞ Distrc(G). The action of elements of Measc(G) on V factors through the tautological map ∞ Measc(G) → Distrc(G), dual to the inclusion C (G) → C(G). NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 35

9.1.2. Recall also that for an irreducible representation ρ of K, we denote by ρ ρ V := ρ ⊗ HomK (V ,V ) the ρ-isotypic component in V . The operator T acts as a projector on V ρ: i.e., its ξρ·µHaarK image is in V ρ, and its restriction to V ρ is the identity. Recall the notation M V K -fin := V ρ; ρ this is the space of K-finite vectors V . Its tautological map into V is injective and has a dense image, called the space of K-finite vectors. We claim: Lemma 9.1.3. For every ρ, the subspace V ∞ ∩ V ρ is dense in V ρ. ρ Proof. Choose a Dirac sequence of smooth functions fn → δ1. Then for every v ∈ V , the elements T ?T (v) ξρ·µHaarK fn·µHaarG ρ ∞ belong to both V and V and converge to v. 9.1.4. A representation V of G is said to be admissible if for every ρ, the vector space V ρ is finite-dimensional. A key observation is: Proposition 9.1.5. Let V be admissible. Then V K -fin ⊂ V ∞. Proof. It is enough to show that V ρ ⊂ V ∞ for every ρ. I.e., we need to show that V ∞∩V ρ equals all of V ρ. However, this subspace is dense by Lemma 9.1.3, and since V ρ is finite-dimesnional, it must be the whole thing. 9.1.6. As a result we obtain that if V is admissible, the Lie algebra g acts on V K -fin. Note that the actions of K and g (on all of V ∞) are compatible in the following sense: (i) For k ∈ K and η ∈ g,

−1 Tk · Tη · Tk = TAdk(η); (ii) The action of k on V ∞ arising from the action of K on V (any vector smooth with respect to G is obviously smooth with respect to K) equals the restriction of the action of g along the tautological map k → g.

9.1.7. We deﬁne the notion of (g,K)-module to be a C-vector space, equipped with an algebraic (i.e., locally ﬁnite) action of K and an action of g that are compatible in the sense that (i) and (ii) above hold.

Let KC be the complex algebraic group corresponding to K. Note that (g,K)-modules can be equivalently deﬁned as complex vector spaces equipped with an algebraic action of KC and an action of gC that are compatible in the similar sense. Thus, (g,K)-modules form a purely algebraic category; we denote it by (g,K)-mod. Let (g,K)-modadm ⊂ (g,K)-mod denote the full subcategory of admissible (g,K)-modules. The assignment V 7→ V K -ﬁn is a functor

(9.1) Rep(G)adm → (g,K)-modadm. 36 DENNIS GAITSGORY

9.1.8. Let M be an admissible (g,K)-module. We can form its algebraic dual M (M ∗)alg := (M ρ)∗, ρ which is equipped with a natural action on g. Equivalently, (M ∗)alg is the subspace of K-ﬁnite vectors in the full linear dual M ∗, the latter being Q (M ρ)∗. ρ 9.2. How well is a representation approximated by its (g,K)-module?

9.2.1. Following Harish-Chandra, we shall sat that two admissible representations V1 and V2 of K -fin K -fin G are infinitesimally equivalent if V1 and V2 are isomorphic as (g,K)-modules.

It is (obviously) not true that if two admissible representations V1 and V2 of G are inﬁnites- imally equivalent then they are isomorphic. For example, take G = K and V1 = L2(K) and V2 = C(K).

The above example may not be too interesting since the representations V1 and V2 are highly reducible. We will bring more interesting examples when we introduce principal series representations. But in case, the phenomenon has to with the fact that a given (g,K)-module can be topologized in many diﬀerent ways so that its completion carries an action of G.

9.2.2. Nonetheless, we have: Theorem 9.2.3.

(a) Let V1 and V2 be two admissible representations of G. Let S : V1 → V2 be a continuous map K -fin K -fin of the underlying vector spaces. Assume that S sends V1 to V2 and that the resulting K -fin K -fin map V1 → V2 is compatible with (g,K)-module structures. Then the initial S was a map of G-representations. K -fin (b) For V ∈ Rep(G)adm and M := V , for every (g,K)-submodule M1 ⊂ M, its closure M1⊂V is a G-subrepresentation. (c) The assignments K -fin (V1 ⊂ V ) 7→ (V1) ⊂ M and M1 7→ M1 ⊂ V define mutually inverse bijections between the set of closed G-subrepresentations of V and the set of (g,K)-submodules of M. Corollary 9.2.4. An admissible representation V is irreducible (i.e., contains no closed G- invariant subspaces) if and only if V K -fin is irreducible as a (g,K)-module. It is of course not true that every G-representation is admissible (take an infinite direct sum of copies of the same representation). But one can ask: is it true that irreducible G- representations are admissible. This was conjectured by Harish-Chandra, but many years later, W. Soergel produced a counterexample. However, we have the following theorem (to be proved next week): Theorem 9.2.5. Every unitary irreducible representation of G is admissible. In Corollary 10.1.6 we will see that a unitary irreducible representation of G is completely determined by its space of K-finite vectors. Theorem 9.2.5 has a converse (also due to Harish-Chandra): NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 37

Theorem 9.2.6. Let M be an irreducible (g,K)-module equipped with an invariant Hermitian positive-deﬁnite form. Then the Hilbert space completion V of M carries a unique unitary G-representation such that V K -ﬁn = M as (g,K)-modules. In the above theorem a form (−, −) is called invariant if

(k · m1, k · m2) = (m1, m2), ∀k ∈ K and (ξ · m1, m2) = −(m1, ξ · m2), ∀ξ ∈ g.

As a formal corollary, one obtains: Corollary 9.2.7. Let M be an admissible (g,K)-module equipped with an invariant Hermitian positive-deﬁnite form. Then the Hilbert space completion V of M carries a unique unitary G-representation such that V K -ﬁn = M as (g,K)-modules.

Proof. By admissibility, M is an orthogonal direct sum of irreducible (g,K)-modules. 9.2.8. The proof of Theorem 9.2.3 is based on the following: Proposition 9.2.9. Let V be an admissible representation of V , and let v be an element of V K -ﬁn. Then for every η ∈ V ∗, the function on G g 7→ η(g(v)) is real analytic. Let us prove Theorem 9.2.3 using this proposition:

K -ﬁn Proof. For point (a), it is enough to show that for v1 ∈ (V1) we have

Tg ◦ S(v1) = S ◦ Tg(v1). ∗ For that it suﬃces to show that for any η ∈ V2 , we have

η(Tg ◦ S(v1)) = η(S ◦ Tg(v1)). Both sides are analytic functions in g. Hence, it is enough to show that all of their derivatives at 1 ∈ G are equal, and that they agree on at least one point on each connected component of G. The ﬁrst condition follows from the compatibility with the action of U(g). The second condition follows from the compatibility with the action of K.

For point (b), we recall that by the Hahn-Banach theorem, the closure of a subspace M1 ⊂ V ⊥ ⊥ ⊥ ∗ equals ((M1) ) , where (M1) ⊂ V is the annihilator of M1. Hence, it suﬃces to show that ⊥ ⊥ for any η ∈ (M1) we have g · η ∈ (M1) , i.e., for every v1 ∈ M1 we have η(g(v1)) = 0. This follows by analyticity in the same way as above. K -ﬁn To prove point (c), we note that for a subrepresentation V1 ⊂ V , the subspace V1 is dense in V1, so its closure is all of V1.

Vice versa, for a sudmodule M1 ⊂ M, it suﬃces to show that for every ρ, the image of T on M lies in M ρ. However, T (M ) ⊂ M ρ, and the assertion follows by ξρ·µHaarK 1 1 ξρ·µHaarK 1 1 continuity. 9.3. Proof of analyticity. The goal of this subsection is to prove Proposition 9.2.9. With no restriction of generality, we can assume that v ∈ V ρ for some ρ ∈ Irrep(K). 38 DENNIS GAITSGORY

9.3.1. Let D be a differential operator of order n on a differentialble (resp., real analytic) manifold X. Let σn(D) be its symbol, i.e., the image of D under the projection ≤n ≤n ≤n−1 n Diff (X) → Diff (X)/ Diff (X) ' SymC∞(X)(Vect(X)). ∗ We can regard σn(D) as a function on T (X), which is homogenous of degree n along the fibers. ∗ Recall that D is said to be elliptic if σn(D)(η) > 0 for all 0 6= η ∈ Tx (X). We have the following basic result (elliptic regularity): Theorem 9.3.2. Let φ be an element of Distr(X) (where Distr(X) is the topological dual of ∞ ∞ Cc (X)). If D(φ) = 0 (i.e., φ(D(f)) = 0 for all f ∈ Cc (X)). Then: (a) The distribution φ is smooth. I.e., it given by a smooth function (times a smooth measure on X). (b) If X is real analytic, and the coefficients of D are real analytic, then φ is real analytic. I.e., it is given by an analytic function (times an analytic measure on X). 9.3.3. We will apply Theorem 9.3.2(b) to X = G and φ being the (continuous) function (times

µHaarG ) given by g 7→ η(g(v)). Thus, our goal is to find an elliptic operator D with analytic coefficients that annihilates this function. We will define D as a left-invariant differential operator corresponding to a certain element u ∈ U(g)≤n. The analyticity is then guaranteed by the construction.

The ellipticity will amount to checking that the image σn(u) of u in U(g)≤n/U(g)≤n−1 ' Symn(g) ∗ satisfies σn(u)(η) > 0 for any 0 6= η ∈ g . 0 0 9.3.4. Write g = g ⊕ zg. Let (−, −) be a bilinear form on g, which is the Killing form on g and a form on zg that is positive-definite on the split part of zg (i.e., zg ∩ p = zg ∩ a) and negative-definite on the compact part of zg (i.e., zg ∩ k). Then (−, −) is the direct sum of a positive-definite form on p and a negative-definite form on k.

9.3.5. Let Cg ∈ Z(g) ⊂ U(g) be the Casimir element corresponding to (−, −). I.e., if ei is an orthonormal basis in g with respect to (−, −), then X 2 Cg = (ei) . i

It is an elementary fact that Cg indeed belongs to Z(g) and is independent of the choice of a basis.

Consider also the element Ck ∈ U(k) ⊂ U(g). Set ue = Cg − 2 · Ck. ∞ ρ ρ Since Z(g) commutes with the action of K, the action of ue on V preserves V . Since V is finite-dimensional, we can find a monic polynomial p (of some degree n) such that p(T ) ue annihilates V ρ. Set u = p(ue). By construction u annihilates v, and hence annihilates our function φ. It remains to check d the ellipticity. Take n = 2d.We have σn(u) = (σ2(ue)) . Hence, it suffices to show that 2 σ2(ue) ∈ Sym (g) NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 39 is elliptic. 0 00 0 Choose an orthonormal bases for g to be of the form {ei}∪{ej }, where {ei} is an orthonormal 00 basis for k, and {ej } is an orthonormal basis for p. Then X 00 2 X 00 2 σ2(ue) = (ej ) − (ei ) . j i I.e., σ2(ue) is the quadratic form corresponding to the bilinear form (−, −)|p − (−, −)|k, and the result follows from the fact that the latter is positive-definite. 9.4. (g,K)-modules vs g-modules. 9.4.1. Consider the forgetful functor (g,K)-mod → g-mod. We can factor it as (g,K)-mod → (g,K0)-mod → g-mod, where K0 is the neutral connected component of K.

Lemma 9.4.2. The functor (g,K0)-mod → g-mod is fully faithful. Its essential image is stable with respect to taking submodules. Proof. The action of g on a vector space determines that of k, and if the latter extends to an action of K0, it does so in a unique way (in this case we say that the action of k integrates to that of K0). This happens of and only if the following happens: (i) The action of the derived Lie algebra k0 is locally ﬁnite; 0 0 (ii) the resulting action of the simply-connected cover of K0 factors through that of K0.

(iii) The abelian algebra zk acts semi-simply with characters given by characters of the torus

(ZK0 )0.

(iv) The two resulting actions of Z 0 ∩ (Z ) agree. K0 K0 0 It is clear that these conditions are stable with respect to taking submodules.

Corollary 9.4.3. If M ∈ (g,K0)-mod is such that the underlying g-module is irreducible, then M itself is irreducible. 9.4.4. Recall that an object M of an abelian category A is said to be finitely generated if for every ascending chain M1 ⊂ M2 ⊂ ... ⊂ M with ∪ Mi = M, we have Mi = M for some i. i We say that A is Noetherian if a subobject of a finitely generated object is finitely generated. If A = A-mod for an associative algebra A, this property is equivalent to A being (left)- Noetherian. In particular, this is the case for A = g-mod, since U(g) is Noetherian (this follows from the fact that gr(U(g)) ' Sym(g) is Noetherian). We say that A is Artinian, if every finitely generated object has finite length. Lemma 9.4.5. The forgetful functor (g,K)-mod → g-mod sends finitely generated objects to finitely generated objects. 40 DENNIS GAITSGORY

Proof. By Lemma 9.4.2, it suﬃces to prove the corresponding fact for the restriction functor (g,K)-mod → (g,K0)-mod. Let M be a ﬁnitely generated (g,K)-module, and let

M1 ⊂ M2 ⊂ ... ⊂ M, ∪ Mi = M i be a chain of (g,K)-submodules. Pick representatives k ∈ K for each element of π0(K). Then each 0 X Mi = k · (Mi) k∈K 0 is a (g,K)-submodule of M. Hence, Mi = M for some i. Pick j large enough so that k · (Mi) ⊂ Mj. Then Mj = M. Corollary 9.4.6. The category (g,K)-mod is Noetherian. 9.4.7. We now claim: Proposition 9.4.8. For an irreducible (g,K)-module, the underlying g-module is a direct sum of finitely many irreducibles. Proof. Let M be an irreducible (g,K)-module. By Lemma 9.4.2, it suffices to show that M is a direct sum of finitely many irreducibles (g,K0)-modules. 0 0 Let M ⊂ M be a maximal (g,K0)-submodule with M/M 6= 0 (such always exists by Zorn’s 0 lemma). Thus, N = M/M is a non-zero irreducible (g,K0)-module. Pick a representative k ∈ K for each element of π0(K). Consider M 00 := ∩ k(M 0) ⊂ M. k Then M 00 is a proper (g,K)-submodule of M, and hence equals zero. Hence, the map M → ⊕ (M 0)k k is injective, where (M 0)k denotes M 0 with the action twisted by the automorphism k. Thus, M, when viewed as a (g,K0)-module is a submodule of a semi-simple module, and hence is semi-simple.

10. Week 5, Day 2 (Thurs, Feb. 23) 10.1. Schur’s lemma for (g,K)-modules.

10.1.1. Here is version of Schur’s lemma for Lie algebras:

Theorem 10.1.2. Let M be an irreducible g-module. Then the map C → Endg(M) is an isomorphism. Proof. Let S be an endomorphism of M. Suppose that it is not a scalar. Consider the map

C[s] → Endg(M), s 7→ S. We claim that it is injective. Indeed, if it was not, it would contain an element of the form Q ni (s − ai) in its kernel, which would mean that one of operators S − ai · Id) is non-invertible. i Since M is irreducible, this would mean that S = ai · Id, contradicting the assumption. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 41

Thus, the above map extends to an (automatically injective) map from the fraction ﬁeld C(s) of C[s] to Endg(M). Note, however, that C(s), when viewed as a vector space over C 1 has dimensional at least continuum: indeed, the elements s−a are all linearly independent. However, we claim that Endg(M) is countable-dimensional. Indeed, for any non-zero vector m ∈ M, evaluation on m deﬁnes an injective map

Endg(M) → M, while M itself is countable-dimensional, being the quotient of U(g) by Ann(m). 10.1.3. Note that the same proof applies for M being an irreducible object in the category (g,K)-mod. We will soon see that any irreducible (g,K)-module is admissible (Theorem 10.3.3). Now, for admissible (g,K)-modules, one can give a simpler proof of Schur: Proposition 10.1.4. Any endomorphism of an irreducible admissible (g,K)-module is a scalar. Proof. It is enough to show that any endomorphism S of an admissible (g,K)-module M has a non-zero eigenspace. Pick ρ such that M ρ is non-zero. Then S preserves M ρ, and since the latter is ﬁnite-dimensional, the assertion follows. 10.1.5. Here is an application of Schur’s lemma. This is a statement complementary to Theo- rem 9.2.5. It says that a unitary irreducible representation is fully determined by the space of its K-ﬁnite vectors, viewed as a (g,K)-module:

Corollary 10.1.6. Let V1 and V2 be two irreducible unitary representations that are inﬁnites- imally equivalent. Then they are isomorphic.

Proof. First off, by Theorem 9.2.5, V1 and V2 are admissible, so we can talk about the corresponding (g,K)-modules. K -fin Set Mi := (Vi) . We can recover Vi as the Hilbert space completion of Mi equipped with the induced Hermitian form. Note that if M is equipped with a (g,K)-invariant Hermitian form then we have a canonical identification M ' (M †)alg, where † means “take dual+complex conjugate”.

Fix an isomorphism S : M1 ' M2. It does not necessarily respect the given inner form, but it does so up to multiplication by a (positive) scalar. Indeed, consider the diagram S M1 −−−−→ M2   ∼ ∼ y y † † alg S † alg ((M1) ) ←−−−− ((M2) ) .

The composite automorphism of M1 respects the (g,K)-action, and hence by Schur’s lemma and Corollary 9.2.4, it is given by multiplication by a scalar. Dividing the initial φ by the square root of this scalar, we can thus assume that φ respects the Hermitian structures. Thus, it deﬁnes an isomorphism of Hilbert spaces V1 → V2. Now the result follows from Theorem 9.2.3(a). 10.2. Action of the center of U(g). 42 DENNIS GAITSGORY

10.2.1. Denote by Z(g) the center of the associative algebra U(g). We can identify Z(g) with invariants in U(g) with respect to the adjoint action of g (or G itself). It is known to be “large”: Consider the filtration on Z(g), induced by the PBW filtration on U(g). ≤n ≤n−1 Lemma 10.2.2. The map from grn(Z(g)) := Z(g) /Z(g) to the space of adg-invariants in grn(U(g)) := U(g)≤n/U(g)≤n−1 ' Symn(g) is an isomorphism. Proof. Follows from complete reducibility of algebraic g-representations: the functor of g- invariants is exact, so sends cokernels to cokernels. Hence, gr(Z(g)) identifies as a commutative algebra with Sym(g)g, and the latter identifies with Sym(h)W , and is known to be isomorphic to be a polynomial algebra of rank equal to the rank of g (i.e., dim(h)). Lifting the generators, we can therefore find a (non-canonical) isomorphism between Z(g) with the same polynomial algebra. However, such isomorphism can be made canonical: this is the Harish-Chandra isomorphism (10.1) Z(g) ' Sym(h)W , to be discussed later. 10.2.3. We claim: Proposition 10.2.4. Let M be an admissible (g,K)-module. Then the action of Z(g) is locally finite. I.e., M splits as a direct sum M ' ⊕ Mχ, such that Z(g) acts on each Mχ be χ∈Spec(Z(g)) a generalized character χ. Proof. Since the action of G (and hence K) commutes with that of Z(g), we obtain that Z(g) preserves each K-isotypic componnent M ρ. Since the latter are finite-dimensional, the assertion follows.

10.2.5. For a given element χ ∈ Spec(Z(g)), let (g,K)-modχ be the full subcategory of (g,K)-mod consisting of modules, on which Z(g) acts with a generalized character χ, i.e., for every m ∈ M there exists a power n such that the ideal (ker(χ))n ⊂ Z(g) annihilates m.

Note that by Schur’s lemma, every irreducible object in (g,K)-mod belongs to (g,K)-modχ, for a uniquely deﬁned χ. We have:

Theorem 10.2.6. The category (g,K)-modχ has only ﬁnitely many isomorphism classes of irreducible objects. Remark 10.2.7. This is a rather deep theorem. Although it is completely algebraic, the initial proof by Harish-Chandra was very indirect and used analysis. Later in the semester we will supply a proof using the localization theory of Beilinson-Bernstein. Assuming the above theorem, we obtain:

Theorem 10.2.8. For an object M ∈ (g,K)-modχ the following conditions are equivalent: (i) M is ﬁnitely generated; (ii) M is of ﬁnite length; (iii) M is admissible. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 43

Proof. The implication (ii) ⇒ (i) is evident. Let us prove that (iii) implies (ii). Let M be an admissible object of (g,K)-modχ, and let

0 = M0 ⊂ M1 ⊂ M2... ⊂ Mn = M be a chain of submodules. We will eﬀectively bound the integer n.

Let Lα be the irreducible objects of (g,K)-modχ; by the second statement in Theorem 10.2.6, ρα there are ﬁnitely many of them. For each α, pick ρα ∈ Irrep(K) so that Lα 6= 0. Let ρ = ⊕ ρα. α

For each i = 1, ..., n there exists an index α so that Lα is a subquotient of Mi/Mi1 . Hence

dim(HomK (ρ, Mi/Mi1 )) ≥ 1. Hence,

dim(HomK (ρ, M)) ≥ n.

Let us show that (i) implies (iii). This follows from the next assertion (of independent interest): Proposition 10.2.9. Let M be a ﬁnitely generated (g,K)-module. Then for any ρ, the isotypic component M ρ is ﬁnitely generated over Z(g).

Proof of Proposition 10.2.9. Let M be a finitely generated (g,K)-module. Then it receives a surjection from a (g,K)-module of the form U(g) ⊗ ρ0 for ρ0 a finite-dimensional representation U(k) of K. Hence, it is enough to show that 0 HomK (ρ, U(g) ⊗ ρ ) U(k) is finitely generated over the center. The latter assertion is enough to check at the associated graded level. I.e., it is enough to check that (Sym(g/k) ⊗ Hom(ρ, ρ0))K is finitely generated as a module over Sym(g)G. Let us identify g with its dual via the Killing form. Denote W := Hom(ρ, ρ0); this is a finite-dimensional K-representation. Thus, we need to show that K (Op ⊗ W ) G G is finitely-generated over Og , where Og maps to Op via the restriction map. By Sect. 8.5.5, the K-orbit of a is dense in p; henve the restriction map under a ⊂ p defines an injection K (Op ⊗ W ) ,→ Oa ⊗ W.

Now the assertion follows from the fact that Oa is finite as a module over g//G. Indeed, a is a closed subvariety in h, and g//G identifies with h//W , and the assertion follows from the fact that h is finite over h//W . 10.3. Some consequences about properties of the category (g,K)-mod. 44 DENNIS GAITSGORY

10.3.1. The implication (i) ⇒ (ii) in Theorem 10.2.8, we obtain:

Corollary 10.3.2. The category M ∈ (g,K)-modχ is Artinian. Combining the implication (ii) ⇒ (iii) in Theorem 10.2.8 with Theorem 10.1.2 we also obtain: Theorem 10.3.3. Every irreducible (g,K)-module is admissible. We note that the proof of Theorem 10.3.3 does not use the (more difficult) theorem Theo- rem 10.2.6. 10.3.4. Combining the above results, we obtain: Corollary 10.3.5. For a (g,K)-module the following conditions are equivalent: (i) M is finitely generated and admissible; (ii) M is finitely generated and its support over Spec(Z(g)) is finite; (iii) M is admissible and its support over Spec(Z(g)) is finite; (iv) M is of finite length. We will refer to (g,K)-modules satisfying the equivalent conditions of Corollary 10.3.5 as Harish-Chandra modules. 10.4. Admissibility of unitary representations. In this subsection we will begin the proof of Theorem 9.2.5, which says that an irreducible unitary representation of G is admissible. We will prove a sharper result:

Theorem 10.4.1. Let V be an irreducible unitary representation of G. Then for any ρ ∈ Irrep(K), we have: dim(V ρ) ≤ dim(ρ)2. 10.4.2. Let V be a topological vector soace. Recall that among the various topologies that exist on the vector space End(V ) of continuous endomorphisms of V , there exists what is called the strong topology: The fundamental system of neighborhoods of 0 in V is given by ﬁnite collections

((v1,U1), ..., (vn,Un)), where vi ∈ V and Ui are neighborhoods of zero in V . The corresponding neighborhood in V consists of those S such that S(vi) ∈ Ui, ∀i = 1, ..., n. Let A be an associative algebra acting on V , i.e., we have a homomorphism A → End(V ). We shall say that V is strongly irreducible if the image of A in End(V ) is dense in the strong topology. 10.4.3. Theorem 10.4.1 follows from the following two statements: Theorem 10.4.4. Let V be an irreducible unitary representation of G. Then V , viewed as acted on by Measc(G), is strongly irreducible. Theorem 10.4.5. Let V be a representation of G, which is strongly irreducible when viewed as acted on by Measc(G). Then for any ρ ∈ Irrep(K), we have: dim(V ρ) ≤ dim(ρ)2. 10.5. Proof of Theorem 10.4.4. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 45

10.5.1. Let H be a Hilbert space. Let A be a subalgebra in End(H) closed under the operation S 7→ S†. Let A be the strong closure of A in End(H). Let Ac ⊂ End(H) be the commutant of A, i.e., Ac = {S0 ∈ End(V ),S0 ◦ S = S ◦ S0 for all S ∈ A}.

We will use the following result of von Neumann: Theorem 10.5.2. The inclusion A ⊂ (Ac)c is an equality.

Corollary 10.5.3. If the inclusion C ,→ Ac is an equality, then the action of A on V is strongly irreducible.

10.5.4. Let A be the image of Measc(G) in End(V ). The operation f(g) 7→ f(g−1) † deﬁnes an involution on C(G), and hence on Measc(G), denoted φ 7→ φ . We have † Tφ† = (Tφ) .

Thus, we obtain that Theorem 10.4.4 follows from the next result, which can be viewed as a Hilbert space version of Schur’s lemma: Theorem 10.5.5. Let V be an irreducible unitary representation of G. Then any endomorphism of V is a scalar. 10.5.6. For the proof of Theorem 10.5.5, we will use the following weak form of the spectral theorem. Let H be a Hilbert space, and S be a continuous self-adjoint endomorphism of H. Then for any λ ∈ R there exists an idempotent πλ of H with the following properties:

(i) Any endomorphism of H that commutes with S commutes also with πλ;

(ii) For any v ∈ Im(πλ) we have (S(v), v) ≤ λ(v, v);

(iii) For any v ∈ ker(πλ) we have (S(v), v) ≥ λ(v, v). 10.5.7. Proof of Theorem 10.5.5. Let S be an endomorphism of V . First oﬀ, we can assume that S is self-adjoint. Indeed, if S is an endomorphism, then so is S†. Then S + S† and i(S − S†) are self-adjoint, and if we can prove that each of them is a scalar, then so is the initial S.

Foe every λ, let πλ be the corresponding idempotent of V . By (i), it commutes with the action of G. Hence, by the irreducibility we either have Im(πλ) = V or πλ = 0.

Let λ0 be the inﬁmum of all λ such that (S(v), v) ≤ λ(v, v); it exists because S is bounded (and non-zero, which we can assume). Then from (ii) we obtain that for all λ < λ0, πλ = 0. Hence, by (iii) we obtain that (S(v), v) ≥ λ(v, v), ∀v ∈ V.

Thus, we obtain that (S(v), v) = λ0(v, v), i.e., S = λ0 · Id. 10.6. Proof of Theorem 10.4.5. 46 DENNIS GAITSGORY

ρ 10.6.1. For ρ ∈ Irrep consider the corresponding projector ξρ on V . Set

Aρ = ξρ · Measc(G) · ξρ.

ρ This is a subalgebra in Measc(G) and it acts on V . It is easy to see that if the action of ρ Measc(G) on V is strongly irreducible, then the action of Aρ on V is strongly irreducible. Theorem 10.4.5 follows from the combination of the following two statements:

Proposition 10.6.2. There exists a family of ﬁnite-dimensional representations πρ of Aρ, such that: (i) Each π is of dimension ≤ n for n = dim(ρ)2;

(ii) For every element a ∈ Aρ there exists a π such that the action of a in π is non-zero. Proposition 10.6.3. Let A be an associative algebra equipped with a family of ﬁnite- dimensional modules satisfying conditions (i) and (ii) from Proposition 10.6.2. Then if V is a topological vector spaced equipped with a strongly irreducible A-action, then dim(V ) ≤ n. In the rest of this subsection we will prove Theorem 10.6.3. 10.6.4. For an associative algebra A and a positive integer r consider the map ⊗r X Pr : A → A, (a1, ..., ar) 7→ (−1)σ · aσ(1) · ... · aσ(n).

σ∈Σr

n 2 Suppose that A = End(C ). It is clear that Pr = 0 for r ≥ n . Let r(n) be the minimal integer such that Pr(n) vanishes. It is easy to see that the function n 7→ r(n) is strictly increasing. Remark 10.6.5. The theorem of Amitzur-Levitzky says that r = 2n.

10.6.6. Let A be as in Proposition 10.6.3 it follows that Pr(n) vanishes on A. Let now V be a topological vector space equipped with a strongly irreducible action of A. By density, we obtain that Pr(n) vanishes also on End(V ). Assume for the sake of contradiction that dim(V ) > n. Then we can split V as a direct sum n+1 0 V = C ⊕ V , where V 0 is some topological vector space. Then End(V ) contains a subalgebra isomorphic n+1 n+1 to End(C ). However, Pr(n) is non-zero on End(C ), since r(n + 1) > r(n). This is a contradiction. 10.7. Proof of Proposition 10.6.2. 10.7.1. Let π be the set of all irreducible finite-dimensional (hence, algebraic) representations of G. Since G is linear (and so admits a faithful finite-dimensional representation), it is clear that for any φ ∈ Measc(G), there exists a π on which it acts non-trivially. ρ Take πρ := π , the ρ-isotypic component in π. We obtain that this family satisfies condition (ii). Hence, it remains to prove the following: Theorem 10.7.2. For every irreducible finite-dimensional representation π of G, the dimension of the space πρ is ≤ dim(ρ)2. The rest of this subsection is devoted to the proof of this theorem. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 47

10.7.3. Consider the corresponding complex groups GC and KC. Let X be the ﬂag variety of GC. We will use the following fundamental result (the Bott-Borel-Weil theorem):

Theorem 10.7.4. Every irreducible representation of GC can be realized as the space of regular sections of a G-equivariant line L bundle on X. 10.7.5. Recall that the Iwasawe decomposition said that K acted transitively on G/P . This implies that the orbit of the unit point under the action of KC on (G/P )C is open. Since M is compact modulo its center, we obtain that the action of KC on X also has an open orbit; denote it X0. The map Γ(X, L) ,→ Γ(X0, L) is an injective map of KC-representations. It remains to show that for any ρ,

0 dim(HomK (ρ, Γ(X , L))) ≤ dim(ρ). C

However, this is true for any KC-homogeneous space equipped with an equivariant line 0 bundle. Indeed, Γ(X , L) is a subrepresentation inside the regular representation Reg(KC) of KC (i.e., the space of regular functions on KC), and ∗ HomK (ρ, Reg(K )) ' ρ , C C by Frobenius reciprocity.

11. Week 6, Day 1 (Thurs, Feb. 28) 11.1. More on the notion of inﬁnitesimal equivalence.

11.1.1. Let V1 and V2 be two admissible G-representations, and let S : M1 → M2 be a map between the underlying (g,K)-modules. It is (obviously) not true that one can continuously extend S to a map from V1 to V2. But one can do so “up to a hut”:

Proposition 11.1.2. There exists a (canonically deﬁned) admissible G-representations V1,2 equipped with maps

S1 S2 V1 ←− V1,2 → V2, such that S1 induces an isomorphism M1,2 → M1, and such that the resulting map

S1 S2 M1 ' M1,2 −→ M2 is the original S. Proof. Consider the graph of S, which is a map

M1 → M1 ⊕ M2

, and compose it with M1 ⊕ M2 → V1 ⊕ V2.

Let V1,2 be the closure of the image of M1 in V1 ⊕ V2. By Theorem 9.2.3, V1,2 is a G- subrepresentation of V1 ⊕ V2, with the required properties. 48 DENNIS GAITSGORY

K×K -fin 11.1.3. Let HG := Distrc(G) be the subspace of Distrc(G) that consists of distributions that are K-finite with respect to both left and right translations. We have K×K -fin Distrc(G) = ⊕ ξρ1 ? Distrc(G) ? ξρ2 , ρ1,ρ2 where the notation ξρ is as in Sect. 1.4.3 (it acts as a projector on the ρ-isotypic component).

The subspace H(G) is closed under convolutions, so it is a subalgebra in Distrc(G). Note, however, that it is non-unital. If V is an admissible G-representation, we have a canonically deﬁned action of H(G) on V K -ﬁn. From Proposition 11.1.2 we obtain:

Corollary 11.1.4. Let V1 and V2 be two admissible G-representations, and let S : M1 → M2 be a map between the underlying (g,K)-modules. Then S intertwines the actions of H(G) on M1 and M2.

Proof. The assertion is obvious when S comes from a map V1 → V2. Now, Proposition 11.1.2 reduces us to this situation. In particular: Corollary 11.1.5. The action of H(G) on V K -fin depends only on the class of infinitesimal equivalence of V . 11.1.6. Let V be an admissible G-representation, and let V ∗ be its dual. Note that (V ∗)K -fin ' (V K -fin)∗,alg.

Denote M := V K -ﬁn. Thus, for m ∈ M and m∗ ∈ M ∗,alg, we obtain the matrix coeﬃcient function MCV,m⊗m∗ g 7→ hm∗, g · m, which is a C∞-function on G From Corollary 11.1.5 we obtain:

Corollary 11.1.7. The function MCV,m⊗m∗ only depends on the infinitesimal equivalence class of V . 11.2. Proof of Theorem 9.2.6. : 11.2.1. Let M be an irreducible admissible (g,K)-module, equipped with a (g,K)-invariant Hermitian form. We want to show that there exists unitary representation of G such that M is its space of K-finite vectors. The proof is based on the following proposition: Proposition 11.2.2. There exists a Banach realization V on M, such that (m, m) ≤ ||m||2. Proof. We will use the fact that any irreducible (g,K)-module admits a Banach realization (to † br proved later). Let V1 be such a realization, and let V1 be its complex-conjugate dual. We have K -fin †,alg † K -fin M ' V1 and M ' (V1 ) . However, the Hermitian form on M gives rise to an identification M ' M †,alg. Thus, we obtain two embeddings † † ı : M → V1 and ı : M → V1 . NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 49

Let V be the closure of the image of M under the diagonal embedding

ı⊕ı† † M → M ⊕ M −→ V1 ⊕ V1 . † Then V is a G-subrepresentation of V1 ⊕ V1 , by Theorem 9.2.3. It is easy to see that it satisﬁes the requirements since (m, m) ≤ ||ı(m)|| · ||ı†(m)|| ≤ (||ı(m)|| + ||ı†(m)||)2.

11.2.3. Let V be a Banach representation, supplied by Proposition 11.2.2. By deﬁnition, the Hermitian form (−, −) extends continuously to V . We claim that it is G-invariant. This would imply the assertion of Theorem 9.2.6 as the desired unitary representation can be deﬁned as the completion of V with respect to (−, −).

11.2.4. To prove the invariance, it suﬃces to show that for m1, m2 ∈ M, the function −1 f(g) = (g · m1, m2) − (m1, g · m2) = 0.

Note that (m1, −) and (−, m2) are continuous functionals on V . Now, by Proposition 9.2.9, it suﬃces to show that all the derivatives of f at 1 ∈ G vanish. However, this follows from the g-invariance of (−, −) on M. 11.3. An algebra that controls (some) (g,K)-modules.

11.3.1. Let ρ be an irreducible K-representation. Set

Mρ := U(g) ⊗ ρ. U(k) This is a (g,K)-module, where we make K act by

k · (u ⊗ m) = Adk(u) ⊗ m. It is is easy to see that ρ Hom(g,K)-mod(Mρ,M) ' M . Set op Aρ := (End(g,K)-mod(Mρ)) . 11.3.2. We have a pair of mutually adjoint functors

Φ: Aρ-mod : (g,K)-mod : Ψ, where Ψ sends M ∈ (g,K)-mod to Hom(g,K)-mod(Mρ,M), viewed as a module over Aρ, and Φ sends Q ∈ Aρ-mod to Mρ ⊗ Q. Aρ

The functor Ψ is exact, and the functor Φ is only right exact. Lemma 11.3.3. The unit of the adjunction Id → Ψ ◦ Φ is an isomorphism. 50 DENNIS GAITSGORY

ρ Proof. For Q ∈ Aρ-mod, since the functor M 7→ M is exact, we have ρ ρ (Mρ ⊗ Q) ' (Mρ) ⊗ Q ' Aρ ⊗ Q ' Q. Aρ Aρ Aρ Corollary 11.3.4. The functor Φ is fully faithful. 11.3.5. We now claim: Proposition 11.3.6. ρ (a) If M is an irreducible (g,K)-module with M 6= 0, then Ψ(M) is an irreducible Aρ-module.

(b) If Q is an irreducible Aρ-module, then Φ(Q) has a unique irreducible quotient, to be denoted MQ. The composite map Q → Ψ ◦ Φ(Q) → Ψ(MQ) is an isomorphism. Proof. For point (a), let Q be a submodule of Ψ(M). By adjunction, we have a non-zero map Φ(Q) → M. Since M is irreducible, the above map is surjective. Since Ψ is exact, the map Q ' Ψ ◦ Φ(Q) → M is surjective. Hence Q is all of Ψ(M). For point (b), by Zorn’s lemma, Φ(Q) admits some irreducible quotient, denote it M. By adjunction, we have a non-zero map Q → Ψ(M). However, by point (a), Ψ(M) is irreducible, so the above map is an isomorphism.

Suppose now that Φ(Q) admits a surjective map to a direct sum M1 ⊕ M2. We claim that one of these modules must be zero. Indeed, applying Ψ, we obtain a (still surjective) map

Q ' Ψ ◦ Φ(Q) → Ψ(M1) ⊕ Ψ(M2).

But Q is irreducible, so one of these maps, say Q → Ψ(M2) must be zero. However, by adjunction, this means that the initial map Φ(Q) → M2 was zero.

11.3.7. Let M be the Levi subgroup of the minimal parabolic Q of G. Denote KM := K ∩ M. This is the maximal compact subgroup in M. Recall that

M = KM × A.

In the next lecture we will construct an injective algebra homomorphism op (11.1) (Aρ) → EndKM (ρ) ⊗ U(a). 11.4. Admissibility of irreducible (g,K)-modules. Our goal on this subsection is to give another proof Theorem 10.3.3, by a method that we will employ again.

11.4.1. Let us decompose ρ into irreducible KM -modules:

⊕ni ρ ' ⊕ πi . i

Let n = max(ni). We will show that any irreducible representation of Aρ is ﬁnite-dimensional of dimension ≤ n. This would prove Theorem 10.3.3 in view of Proposition 11.3.6. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 51

11.4.2. Recall the notations of Sect. 10.6.4. Since a is commutative, the polynomial Pr(n) vanishes on EndKM (ρ) ⊗ U(a). The existence of the homomorphism (11.1) implies that Pr(n) vanishes on Aρ. We now claim:

Proposition 11.4.3. Let A be an associative algebra such that Pr(n) vanishes on A. Then any irreducible representation of A is ﬁnite-dimensional of dimension ≤ n. Proof. We claim that if M is an A module that contains n + 1 linearly independent vectors, then A has a subquotient isomorphic to Matn+1,n+1. This would be a contradiction since Pr(n) is non-zero on Matn+1,n+1. The existence of a subquotient follows from Burnside’s theorem: if M is an irreducible 0 0 A-module, and m1, ..., mk ∈ M linearly independent vectors, and m1, ..., mk some k-tuple of vectors, then there exists an element a ∈ A such that 0 a · mi = mi, i = 1, ..., k. 11.5. Construction of the homomorphism (11.1).

11.5.1. Consider the tensor product

Fρ := U(m) ⊗ Mρ, U(q) which we can also think of as C ⊗ Mρ. U(n)

It is acted on by m via the left action on the ﬁrst factor. Moreover, it is easy to see as in Sect. 11.3.1 that Fρ is naturally an (m,KM )-module. op In addition, Fρ carries a commuting action of (Aρ) via the second factor.

11.5.2. Using the g = a ⊕ n ⊕ k decomposition, we can write

Fρ := U(m) ⊗ U(g) ⊗ ρ ' U(m) ⊗ U(q) ⊗ ρ ' U(m) ⊗ ρ U(q) U(k) U(q) U(kM ) U(kM ) as an (m,KM )-module. So, this is the object of the same nature as Mρ but for the pair (m,KM ) and the KM -representation ρ. Further, writing U(m) (as an algebra) as

U(m) ' U(a) ⊗ U(kM ), we obtain that Fρ is isomorphic to U(a) ⊗ ρ, as an (m,KM )-module, where m ' a ⊕ kM acts via the action of a on the ﬁrst factor and kM on the second factor.

op 11.5.3. Thus, we obtain that the action of (Aρ) on Fρ gives rise to a map op (Aρ) → End(m,K)-mod(Fρ) ' U(a) ⊗ EndKM (ρ). This is the desired map (11.1). Let us now prove that it is injective. 52 DENNIS GAITSGORY

K 11.5.4. Consider the algebra U(g) of AdK -invariants in U(g). We claim that it naturally acts on Mρ on the right by (g,K)-module endomorphisms. Indeed, this action is given by (u ⊗ v) · u0 7→ u · u0 ⊗ v.

Hence, we obtain an algebra map K U(g) → Aρ.

Let Iρ be the kernel of the action map U(k) → End(ρ). This is a two-sided ideal in U(k). Consider the left ideal U(g) · Iρ ⊂ U(g). Note, however, that the intersection K Jρ := U(g) ∩ U(g) · Iρ is a two-sided ideal in U(g)K . K It is easy to see that the above map U(g) → Aρ factors through a map K (11.2) U(g) /Jρ → Aρ. Proposition 11.5.5. (a) The map (11.2) is an isomorphism. (b) The composite map K op op (U(g) /Jρ) → (Aρ) → U(a) ⊗ EndKM (ρ) is injective. Clearly, Proposition 11.5.5 implies that the map (11.1) is injective. Proof of Proposition 11.5.5. Note that all objects in sight carry a filtration, induced by the canonical filtration on U(g). To prove the proposition, it suffices to show that both (a) and (b) hold at the associated graded level. We have: K K K K gr(U(g) /Jρ) ' (gr(U(g) ⊗ U(k)/Iρ)) ' (gr(U(g) ⊗ End(ρ))) ' (Sym(g/k) ⊗ End(ρ)) U(k) U(k) and K K K gr Aρ ' gr(Hom(ρ, U(g) ⊗ ρ) ) ' (gr(Hom(ρ, U(g) ⊗ ρ)) ' (Sym(g/k) ⊗ End(ρ)) , U(k) U(k) and it is easy to see that under these identifications, the associated graded of (11.2) is the identity map on (Sym(g/k) ⊗ End(ρ))K . This proves point (a). Similarly, the associated graded of the map (11.1) is the map (11.3) (Sym(g/k) ⊗ End(ρ))K → (Sym(a) ⊗ End(ρ))KM , where the map g/k → a is g/k ' q/kM → m/kM ' a. Thus, it remains to see that (11.3) is injective. Identifying g with its dual via the Killing form and denoting W := End(ρ), interpret the above map as

K KM (Op ⊗ W ) → (Oa ⊗ W ) , given by restriction along a ,→ p. Now, the assertion follows from the fact that under the adjoint action of K on p, the orbit of a is dense in p, see Sect. 8.5.5. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 53

12. Week 6, Day 2 (Thurs, March 2) 12.1. Induced representations.

12.1.1. Let Q0 be a parabolic subgroup of G; let N 0 denote its unipotent radical, and let M 0 denote its Levi quotient. Let W be a representation of M 0, which we regard as a representation of Q0.

G We deﬁne the induced representation IQ0 (W ) as follows: its space consists of continuous functions f : G → W that satisfy f(g · q) = q−1 · f(g), g ∈ G, q ∈ Q0. Equivalently, we can identify this space with the space of functions f : G/N 0 → W that satisfy f(g · m) = m−1 · f(g), g ∈ G, m ∈ M 0, since Q0 acts on W via M 0. Frobenius reciprocity implies: Lemma 12.1.2. For a G-representation V , the space of continuous maps of G-representations G 0 V → IQ0 (W ) identiﬁes with the space of maps of Q -representations V → W .

G 12.1.3. Let us identify what IQ0 (W ) looks like as a K-representation. 0 0 0 0 From the fact that G = K · Q and K ∩ Q = K ∩ M =: KM 0 is the maxmal compact in M , G we obtain that the restriction of IQ0 (W ) identiﬁes with the space of functions −1 f : K → W such that f(k · m) = m · f(k), k ∈ G, m ∈ KM 0 , i.e., we have a canonical isomorphism

G G K M 0 (12.1) Res ◦ I 0 ' I ◦ Res K Q KM0 KM0

G This point of view lets us endow IQ0 (W ) with a variety of diﬀerent topologies, and pass to the corresponding completions. For example if W is ﬁnite-dimensional, we can consider the Lp topology on (12.1). It is easy to see that the action of G will still be continuous (but it will not preserve the Lp norm; only the action of K does).

12.1.4. We claim:

G Proposition 12.1.5. Suppose that W is admissible. Then IQ0 (W ) is admissible. Proof. By (12.1) For a ﬁnite-dimensional representation ρ of K, we have

G 0 HomK (ρ, IQ (W )) ' HomKM0 (ρ, W ). 12.2. Induction for (g,K)-modules. 54 DENNIS GAITSGORY

12.2.1. We will now describe the algebraic counterpart of the induction construction. It will 0 use D-modules. Consider the category (m ,KM 0 )-mod. We have a functor G 0 rM 0 :(g,K)-mod → (m ,KM 0 )-mod,M 7→ Mn0 , where the subscript n0 means taking coinvariants with respect to the Lie algebra n0. The result 0 0 0 carries an action of (m ,KM ) because M normalizes n . G We can also rewrite rM 0 as M 7→ U(m0) ⊗ M. U(q0) It is called the Jacquet functor.

G G 12.2.2. The functor rM 0 admits a right adjoint by general nonsense. We will denote it by iQ0 . We observe: Proposition 12.2.3. The natural transformations 0 (g,K) G K (m ,KM0 ) Res ◦ i 0 → Ind ◦ Res K Q KM0 KM0 and 0 (m ,KM0 ) K G (g,K) coInd ◦ Res → r 0 ◦ coInd , KM0 KM0 Q K induced by the tautological natural transformation 0 K (g,K) (m ,KM0 ) G Res ◦ Res → Res ◦ r 0 KM0 K KM0 M are isomorphisms.

G In other words, whatever the functor iQ0 happens to be, we know what it does at the level of K-representations: this is just induction from KM 0 to K, i.e., the following diagram commutes:

iG Q0 0 (g,K)-mod ←−−−− (m ,KM 0 )-mod

  (m0,K ) (g,K)  M0 Res ResK K y y M0

K-mod ←−−−− KM 0 -mod. Note this is an algebraic counterpart of (12.1). Proof. It suﬃces to prove the second isomorphism, as the ﬁrst one is obtained by passing to right adjoints. (g,K) For the second isomorphism, we note that the functor coIndK is given by ρ 7→ U(g) ⊗ ρ U(k)

Using the fact that 0 0 k ⊕ q g and q ∩ k = kM 0 , we obtain: U(m0) ⊗ U(g) ⊗ ρ ' U(m0) ⊗ U(q0) ⊗ ρ ' U(m0) ⊗ ρ, 0 0 U(q ) U(k) U(q ) U(kM0 ) U(k) as desired. G Corollary 12.2.4. The functor iQ0 preserves admissibility. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 55

0 0 12.2.5. Let W be an admissible M -representation. Let L denote the underlying (m ,KM 0 )- module. We claim: G G Proposition 12.2.6. The (g,K)-module underlying IQ0 (W ) identifies canonically with iQ0 (L). Proof. We first construct a map of (g,K)-modules G K -fin G (12.2) (IQ0 (W )) → iQ0 (L). G By the definition of iQ0 (L), the datum of such a map is equivalent to that of a map of 0 (q ,KM 0 )-modules G K -fin (12.3) (IQ0 (W )) → L. By Frobenius reciprocity (i.e., Lemma 12.1.2) we have a map of Q0-representations G IQ0 (W ) → W.

G K -fin KM0 -fin Under this map the subspace IQ0 (W )) gets sent to W = L. This defines the 0 desired map (12.3). It is easy to see that it indeed respect the (q ,KM 0 )-action. Thus, by adjunction we also obtain the map (12.2). In order to see that this map is an isomorphism, it suffices to show that it is such at the level of K-representations. We have: (g,K) G K -fin G G K -fin ResK ((IQ0 (W )) ) ' (ResK (IQ0 (W ))) , which by (12.1) identifies with 0 (IK ◦ ResM (W ))K -fin, KM0 KM0 K and the latter is easily seen to identify with Ind 0 (L). KM Now, (g,K) G K Res (i 0 (L)) ' Ind 0 (L) K Q KM by Proposition 12.2.3. K By unwinding the definitions, it is easy to see that the resulting map from Ind 0 (L) to itself KM is the identity. 12.3. Explicit description of the induction functor on (g,K)-modules. We will now G describe the functor iQ0 explicitly. In doing so we will used modules over algebraic differential operators, a.k.a., D-modules. For the duration of this subsection we be in the context of algebraic geometry, so will write G, K, etc, for their complexifications. 0 0 0 12.3.1. Recall that the K-orbit of 1 in X := G/Q is open; denote it by X0. This is an affine scheme. Denote Xe 0 := G/N 0; 0 0 0 0 0 this is a M -torsor over X . Let Xe0 be the preimage of X0 in Xe0. This is also an affine scheme. The scheme Xe 0 carries an action of G (in particular K) by left translations, and a commuting action of M 0 by right translations. 0 0 Consider the ring of differential operators D(Xe0). The above action of G on Xe gives rise to an algebra homomorphism l 0 0 ι : U(g) → D(Xe ) ⊂ D(Xe0) 56 DENNIS GAITSGORY and the right action of M 0 gives rise to an algebra homomorphism

r 0 0 0 ι : U(m ) → D(Xe ) ⊂ D(Xe0). The images of these two algebras commute with each other. In particular, every D-module 0 on Xe0 carries a left action of g and a right-action of m.

0 12.3.2. We let Y be the closed subscheme of Xe0 equal to the orbit of 1 under the K-action; note Y is isomorphic to K as K ∩ N 0 = {1}.

0 We will now consider a particular D-module on Xe0, denoted it by δ 0 , to be thought of as Y,Xe0 2 0 the δ-distribution on Y inside Xe0.

0 Explicitly, δ 0 is obtained by quotienting D(Xe0) by the left ideal gererated by: Y,Xe0 0 0 (i) IY ⊂ O 0 ⊂ D(Xe0), where IY is the ideal of Y in Xe0; Xe0 l 0 (ii) ι (k) ⊂ D(Xe0). Note that since Y is invariant under left translations by K, the elements from ιl(k) normalize the ideal IY .

12.3.3. We claim that the left action of g on δ 0 extends to a structure of (g,K)-module, and Y,Xe0 0 0 the above action of m extends to a structure of (m ,KM 0 )-module.

0 0 Indeed, the action of K on Xe0 deﬁnes an action of K on D(Xe0). The derivative of this action is given by 0 l l ξ ∈ k, d ∈ D(Xe0) 7→ ι (ξ) · d − d · ι (ξ).

l This action of K preserves the ideal deﬁning δ 0 . Now, if ξ ∈ k, then the element ι (ξ) Y,Xe0 belongs to this ideal and so the above action of ξ ∈ k on δ 0 is given by left multiplication by Y,Xe0 ιl(ξ).

0 0 0 0 Similarly, the right action of M on Xe0 deﬁnes an action of M on D(Xe0). The derivative of this action is given by 0 0 r r ξ ∈ m , d ∈ D(Xe0) 7→ ι (ξ) · d − d · ι (ξ).

The induced action of KM 0 preserves the ideal deﬁning δ 0 . Further, we claim that elements Y,Xe0 r of the form ι (ξ) for ξ ∈ kM 0 belong to this ideal. Indeed, this follows from the fact that the restriction of the vector ﬁeld ιr(ξ) to Y is of the form

X l fi · ι (ξi), fi ∈ OY , ξi ∈ kM 0 . i

r Hence, the above action of ξ ∈ kM 0 on δ 0 is given by left multiplication by ι (ξ). Y,Xe0

2 0 In the D-module language, it is the direct image under Y,→ Xe0 of the D-module OY tensored with the line top ⊗−1 (Λ (k/kM0 )) . NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 57

12.3.4. In what follows we will need the following construction. Let M1 and M2 be a right and a left (g,K)-modules respectively. In this case we can form the vector space3

K K M1 ⊗ M2 := (M1 ⊗ M2) . g/k U(g)

Note that when g = k, the projection

K K (M1 ⊗ M2) → M1 ⊗ M2 k is an isomorphism (this can be seen by identifying K-invariants with K-coinvariants).

So, we should think of the functor

K M1,M2 7→ M1 ⊗ M2 g/k as the operation of taking invariants in the K-direction and coinvariants with respect to g/k.

For example, if (g,K) M1(ρ) = coIndK := ρ ⊗ U(g), U(k) we have

K K (12.4) M1 ⊗ M2 ' (ρ ⊗ M2) . g/k

Remark 12.3.5. In the particular case when G is compact modulo its center (a situation that will be of particular interest for us, because we will take G = M, the Levi of the minimal K parabolic), the operation ⊗ can be described in simpler terms. Namely, g/k

K K M1 ⊗ M2 ' (M1 ⊗ M2) . g/k U(a)

0 12.3.6. We take the module δ 0 , and given L ∈ (m ,KM 0 )-mod we form Y,Xe0

K 0 0 G M iQ0 (L) := δ 0 ⊗ L. Y,Xe0 0 m /kM0

The action of g via ιl makes it into an object of (g,K)-mod. We will prove:

Theorem 12.3.7. There is a canonical isomorphism of functors

G 0 G 0 iQ0 ' iQ0 , (m ,KM 0 )-mod → (g,K)-mod. 12.4. Proof of Theorem 12.3.7.

3The deﬁnition given below is ﬁne when K is reductive, and when we are interested in the non-derived functor. In general, more care is needed. 58 DENNIS GAITSGORY

12.4.1. We ﬁrst claim that there exists a canonical isomorphism of functors 0 (g,K) 0 G K (m ,KM0 ) (12.5) Res ◦ i 0 → Ind ◦ Res , K Q KM0 KM0 0 G i.e., that at the level of K-representations, iQ0 does the right thing.

Indeed, we claim that δ 0 when regarded as a K-module with respect to the left action Y,Xe0 0 and a (m ,KM 0 )-module with respect to the right action, identiﬁes canonically with

OY ⊗ U(m). U(kM0 )

This would imply the isomorphism (12.5) in view of (12.4), since Y ' K as a K × KM 0 - scheme.

To establish the desired description of δ 0 we note that there is a canonical map Y,Xe0

OY → δ 0 Y,Xe0 as K × KM 0 -modules. By adjunction, this gives rise to a map

OY ⊗ U(m) → δ 0 Y,Xe0 U(kM0 ) 0 as K-modules and (m ,KM 0 )-modules. To check that this map is an isomorphism, it is enough to do so at the associate graded level. The latter, however, is easily seen to be the identity map on 0 OY ⊗ Sym(m /kM 0 ). 0 0 G 12.4.2. We next consider the case of Q = G. We claim that iG is indeed isomorphic to the identity functor on (g,K)-mod. I.e., we claim: Proposition 12.4.3. The functor K M 7→ δK,G ⊗ M g/k is isomorphic to the identity functor on (g,K)-mod. Proof. For M ∈ (g,K)-mod, the action of K on M gives rise to a map

M → OK ⊗ M. K Its image lies in (OK ⊗ M) , where we consider the diagonal action of K comprised from the given action on M and the action of K on OK by right translations. Composing, we obtain a map K K K M → (OK ⊗ M) → (δK,G ⊗ M) → δK,G ⊗ M. g/k It is straightforward to check that the above map is a map of (g,K)-modules. This deﬁnes a natural transformation 0 G Id → iG. To check that it is an isomorphism, it suﬃces to show that the induced natural transformation (g,K) (g,K) 0 G ResK → ResK ◦ iG is an isomorphism. However, it is straightforward to check that the above map is the isomorphism of (12.5) in the particular case of Q0 = G. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 59

12.4.4. Next we construct a natural transformation G 0 G (12.6) rQ0 ◦ iQ0 → Id . 0 In view of Proposition 12.4.3, this amounts to constructing a map of (q ,KM 0 )-modules

δ 0 → δK 0 ,M 0 Y,Xe0 M 0 0 that respects the (q ,KM 0 )-action on the left and the (m ,KM 0 )-action on the right. 0 0 0 0 Consider M as a subscheme in Xe0 (i.e., the ﬁber over 1 of G/N → G/Q ) and consider the tensor product

OM 0 ⊗ δ 0 . Y,Xe0 O 0 Xf0 This is a D-module on M 0. The projection

δ 0 → OM 0 ⊗ δ 0 Y,Xe0 Y,Xe0 O 0 Xf0 0 0 respects the (q ,KM 0 )-action on the left and the (m ,KM 0 )-action on the right. Thus, it suﬃces to construct an isomorphism of D-modules on M 0

δK 0 ,M 0 ' OM 0 ⊗ δ 0 . M Y,Xe0 O 0 Xf0 We construct a map

D(M) → OM 0 ⊗ δ 0 Y,Xe0 O 0 Xf0 by sending the generator to 1 ∈ OM 0 . It is easy to see that this map factors via a map

(12.7) δK 0 ,M 0 → OM 0 ⊗ δ 0 . M Y,Xe0 O 0 Xf0 To show that the latter map is an isomorphism, we do it at the associated graded level. However, it is easy to see that gr of both sides identiﬁes naturally with

0 OKM0 ⊗ Sym(m/kM ) so that (12.7) induces the identity map. 12.4.5. By adjunction, the map (12.6) constructed above, gives rise to a natural transformation 0 G G iQ0 → iQ0 . We claim that this natural transformation is an isomorphism. To prove this, it is sufficient to show that the induced natural transformation (g,K) 0 G (g,K) G ResK ◦ iQ0 → ResK ◦ iQ0 is an isomorphism. 0 However, by (12.5), the left-hand side identifies with IndK ◦Res(m ,KM0 ), and the right-hand KM0 KM0 side identifies with the same by Proposition 12.2.3. Now, by unwinding the constructions, we see that the resulting natural transformation from 0 IndK ◦ Res(m ,KM0 ) to itself is the identity map. KM0 KM0 60 DENNIS GAITSGORY

13. Week 7, Day 1 (Tue., March 7) 13.1. Harish-Chandra homomorphism. 13.1.1. Let Q0 be a parabolic in G. Consider the tensor product F := U(m0) ⊗ U(g), U(p0) as a right g-module and a left m0-module. We claim: Proposition 13.1.2. The action of Z(g) ⊂ U(g) on F factors through a uniquely defined map φ : Z(g) → Z(m0) and the action of Z(m0) ⊂ U(m). Furthermore, φ is injective and Z(m0) is finite as a Z(g)- module. Remark 13.1.3. The above map is most commonly used when Q0 = B, and so m0 = h. It is usually referred to as the Harish-Chandra homomorphism. Proof of Proposition 13.1.2. An element z ∈ Z(g) defines an endomorphism of F as a g-module and as a m0-module. We claim that for any endomorphism S of F as an g-module, the image S(1) of the element 1 ∈ F lies in U(m0) ' U(m0) ⊗ U(p0) ⊂ U(m0) ⊗ (U(p0) ⊗ U(n0−)) ' U(m0) ⊗ U(g) = F. U(p0) U(p0) U(p0) 0 Indeed, S(1) is invariant under the adjoint action of zM 0 ⊂ m , but the subspace of such elements in U(n0−) equals C. Hence 1 · z = φ(z) · 1 for a well-defined element φ(z) ∈ U(m). For u ∈ U(m0) we have u · φ(z) = u · z = z · u = φ(z) · u, (the second equality since z ∈ Z(g)), as elements in U(m0) ⊂ F . Hence, φ(z) ∈ Z(m0). A similar argument shows that φ is an algebra homomorphism. To show that φ injective and Z(m0) is finite as a Z(g)-module it is enough to do so at the associated graded level. By transitivity, it is enough to consider the case of Q0 = B. The corresponding map is Sym(g)G → Sym(g/n)T ' Sym(h). Identifying g with its dual via the Killing form, the latter map is the restriction map Sym(g)G → Sym(h), which is injective because the orbit if h in g under the adjoint action is dense. G 13.1.4. Consider the functor rQ0 . We claim: G Lemma 13.1.5. For M ∈ (g,K)-mod, the action of Z(g) on rQ0 (M) induced by the Z(g)-action 0 0 G on M equals the action obtained from φ : Z(g) → Z(m ) and the action of Z(m ) on rQ0 (M) as an m0-module. 0 G Proof. Follows from the fact that the functor g-mod → m -mod underlying rQ0 is given by M 7→ F ⊗ M. U(g) NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 61

By adjunction, we obtain:

0 G Corollary 13.1.6. For L ∈ (m ,KM 0 )-mod, the action of Z(g) on iQ0 (L) as a (g,K)-module equals the action induced by the action of Z(g) on L via φ. 13.1.7. We claim: Proposition 13.1.8. G (a) The functor rM 0 sends finitely generated objects in (g,K)-mod to finitely generated objects in (m,M 0)-mod. G (b) The functor rM 0 sends objects of finite length in (g,K)-mod to objects of finite length in (m,M 0)-mod. Proof. Point (a) follows from the second isomorphism in Proposition 12.2.3. Point (b) follows from point (a) and Corollary 10.3.5. Remark 13.1.9. The proof of Proposition 13.1.8 used Corollary 10.3.5, and that in turned used the (non-trivial) Theorem 10.2.6, applied to the reductive group M 0. However, when Q0 is the minimal parabolic Q, so that the Levi M 0 = M is compact modulo its center, the conclusion of Theorem 10.2.6 is evident. G So, we obtain that the corresponding functor rQ sends (g,K)-modules of finite length to finite-dimensional (m,KM )-modules.

The proof amounts to the fact that for M ∈ (g,K)-mod of finite length, the (m,KM )-module G rM (M) is finitely generated and has finite support over Z(g). This implies that its support over Z(m) is also finite. The latter means that its support in Spec(U(a)) is finite and that it has only finitely many KM -isotypics components. 13.2. The subquotient theorem.

13.2.1. In this subsection we will prove the following fundamental result, known as the Sub- quotient Theorem:

G Theorem 13.2.2. Every irreducible (g,K)-module can be realized as a subquotient of iP (L), where P is the minimal parabolic and L is a finite-dimensional (m,KM )-module. The particular significance of this theorem is explained by the following corollary: Theorem 13.2.3. Every irreducible (g,K)-module can be realized as the (g,K)-module underlying an admissible representation of G on a Banach (and even Hilbert) space. Let us see how Theorem 13.2.2 implies Theorem 13.2.3: Proof of Theorem 13.2.3. Let us realize a given irreducible (g,K)-module M as a subquotient G G G of iQ(L). By Proposition 12.2.6 we can realize iQ(L) as the space of K-finite vectors in IQ (L), G which is a Banach representation of G (and also in the L2-version of IQ (L), which is a Hilbert (but not unitary!) representation of G). G Hence, by Theorem 9.2.3, M can be realized as a subquotient of IQ (L) (or its L2 version). Remark 13.2.4. Note that Theorem 13.2.2 does not explicitly mention the fact that the (g,K)- G modules iQ(L) have a finite length. The latter is true, but is a hard theorem, essentially equivalent to Theorem 10.2.6, see below. 62 DENNIS GAITSGORY

13.2.5. Let us make the following observation on the structure of the theory: Proposition 13.2.6. The following assertions are logically equivalent: G (i) For an irreducible (m,KM )-module L, the (g,K)-module iQ(L) is of finite length. G (i’) For a finite-dimensional (m,KM )-module L, the (g,K)-module iQ(L) is finitely generated. (ii) The assertion of Theorem 10.2.6 holds, i.e., for every character χ of Z(g) there are only finitely many classes of irreducible objects in (g,K)-modχ. Proof. Let us show that (i) implies (ii). By Theorem 13.2.2, we know that every irreducible G (g,K)-module can be realized as a subquotient of iQ(L) for some irreducible (m,KM )-module G L. By Corollary 13.1.6, if L is irreducible, then Z(g) acts on iQ(L) by a single character, obtained via the Harish-Chandra homomorphism φ : Z(g) → Z(g) from the character by which Z(m) acts on L.

Hence, it suffices to show that, given χ, there are only finitely many irreducible (m,KM )- modules L, on which Z(g) acts by χ. Since the map Spec(Z(m)) → Spec(Z(g)) is finite, it suffices to show that for a given character χ0 of Z(m), there are at most finitely many irreducible 0 (m,KM )-modules L, on which Z(m) acts by χ . In fact, there is at most one such module: Te datum of χ0 determines the action of a, and it is known that finite-dimensional represen- 4 tations of a compact group (in our case KM ) are distinguished by the action of the center of the universal enveloping algebra. Let us show that (ii) implies (i). We have seen that Theorem 10.2.6 implies Corollary 10.3.5. G In particular, in order to prove that iQ(L) is of finite length it is sufficient to it is admissible (which follows from Corollary 12.2.4) and that its support over Z(g) is finite (which follows from Corollary 13.1.6). Clearly (i) implies (i’). For the inverse implication we will use the fact (to be proved next G G ∗ ∗ time) that the algebraic dual of iQ(L) is isomorphic to iQ(L ), where L is the dual of L (up to a ρ-shift). If ... ⊂ Mi ⊂ Mi+1 ⊂ ... G is an infinite chain of subobjects of iQ(L) (with non-zero subquotients), consider the corresponding chain ⊥ ⊥ ... ⊂ (Mi+1) ⊂ (Mi) ⊂ ... G ∗ in iQ(L ). Assuming (i’) and using the fact that the category (g,K)-mod is Noetherian, we obtain that both these chains stabilize on the right. This is a contradiction. 13.3. Proof of the subquotient theorem.

13.3.1. Let ρ be an irreducible representation of K. Recall the algebra Aρ from Sect. 11.3. We claim that it suﬃces to show that every irreducible Aρ-module can be realized as a subquotient G G of Ψ(iQ(L)) = HomK (ρ, iQ(L)) for some ﬁnite-dimensional representation L of M. Indeed, let M be an irreducible (g,K)-module. Let ρ be such that Mρ 6= 0. Denote Q := G Ψ(M). Let L be such that Q can be realized as a subquotient of Ψ(iQ(L)). I.e., we have submodules G Q1 ⊂ Q2 ⊂ Ψ(iQ(L)) and an isomorphism Q ' Q2/Q1.

4If we just consider the action of the Casimir, for every eigenvalue there will be at most ﬁnitely many irreducibles on which it acts with this eigenvalue. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 63

Consider the corresponding maps

G Φ(Q1) → Φ(Q2) → iQ(L). We claim that the composition

G G Φ(Q2) → iQ(L) → coker(Φ(Q1) → iQ(L)) is non-zero. Indeed, if it were, then applying the functor Ψ we would obtain that the composition

G G Ψ ◦ Φ(Q2) → Ψ(iQ(L)) → coker(Ψ ◦ Φ(Q1) → Ψ(iQ(L))) is zero. However, since Ψ ◦ Φ ' Id, the latter map identiﬁes with

G G Q2 → Ψ(iQ(L)) → coker(Q1 → Ψ(iQ(L))), which was non-zero by assumption. Therefore, since Φ is right exact, we obtain that Φ(Q) surjects onto a subquotient, (denote 0 G 0 it M ) of iQ(L). We claim that in this case M is a quotient of M . Indeed, this follows from the fact that Φ(Q) has a unique irreducible quotient (Proposition 11.3.6(b)).

G 13.3.2. Let us now write down explicitly the functor Ψ ◦ iQ. (This will be a lot easier than G writing down iQ itself).

G G By deﬁnition, the functor Ψ ◦ iQ is the right adjoint of the functor rQ ◦ Φ. Now, the latter functor sends

Q 7→ U(m) ⊗ U(g) ⊗ ρ ⊗ Q ' Fρ ⊗ Q, U(p) U(k) Aρ Aρ where we regard Fρ as an (m,KM )-module equipped with a commuting right action of Aρ. G Note that Fρ, veiwed as an (m,KM )-module identiﬁes with rQ(U(g) ⊗ ρ). U(k)

G Hence, the functor Ψ ◦ iQ is given by

L 7→ Hom(m,KM )-mod(Fρ, L).

Recall that Fρ, when viewed as an (m,KM )-module, isomorphic to U(m) ⊗ ρ. U(kM )

G So, the composition of Ψ ◦ iQ with the forgetful functor Aρ-mod → Vect is just

L 7→ HomKM (ρ, L).

We also note:

Lemma 13.3.3. The module Fρ are ﬁnitely generated over Z(g).

Proof. Note that by Lemma 13.1.5 the Z(g)-action on Fρ, equals the action obtained from φ : Z(g) → Z(m) and the Z(m) action on Fρ as a (m,KM )-module. Clearly, Fρ is ﬁnitely generated as a module over U(a), and hence over all of Z(m). Now the assertion follows from the fact that Z(m) is ﬁnitely generated as a module over Z(g). 64 DENNIS GAITSGORY

13.3.4. Since the (right) action of Aρ on Fρ is faithful by Sect. 11.5, (and Fρ is ﬁnitely generated over U(a)), we can ﬁnd an embedding ⊕k (13.1) Aρ ,→ Fρ as right Aρ-modules, for some k.

Let Q be an irreducible (automatically, finite-dimensional) module over Aρ; consider its linear ∗ dual Q as a right Aρ-module; it is irreducible. In particular, it can be realized as a quotient of ⊕k Aρ. It suffices to show that Q can be realized as a subquotient of Hom(m,KM )-mod(Fρ , L) for some L. Let χ be the character of Z(g) through which it acts on Q. Then Z(g) acts by the same ∗ character on Q . Let Iχ ⊂ Z(g) be the corresponding ideal. Recall that by Lemma 13.3.3, Fρ is finitely generated as a Z(g)-module. It follows from Artin-Rees applied to the embedding 0 ⊕k n (13.1) that Aρ/Iχ is a subquotient of F := Fρ /Iχ for some n. 0 This F is an (m,KM )-module that carries a commuting right action of Aρ. It suffices to show 0 that Q can be realized as a subquotient of Hom(m,KM )-mod(F , L) for some finite-dimensional (m,KM )-module L. Note that F 0 is finite-dimensional as a vector space. Take L = F 0 ⊗ (F 0)∗, considered as a module over (m,KM ) via the action on the first factor. We claim that it does the job. Indeed, the evaluation map F 0 ⊗ (F 0)∗ → C defines a surjection 0 0 ∗ Hom(m,M)(F , L) → (F ) 0 ∗ as left Aρ-modules. However, (F ) contains Q as a subquotient, by construction.

14. Week 7, Day 2 (Tue., March 9) 14.1. Normalized induction and duality on induced representations. 14.1.1. Consider the adjoint action of q0 on q0, and let us consider the top exterior power of this action. We obtain an (algebraic) character of the group Q0, i.e., a homomorphism 0 Q → Gm, which automatically factors via a homomorphism 0 (14.1) M → Gm. In its turn, the homomorphism (14.1) factors through a homomorphism 0 0 0 (14.2) M /[M ,M ] → Gm. Note that M 0/[M 0,M 0] is a connected torus deﬁned over R. Therefore it (rather, the group of its R-points) can be written as

KM 0/[M 0,M 0] × AM 0/[M 0,M 0]. 0 (Note that when Q is the minimal parabolic Q, then AM 0/[M 0,M 0] ' A.)

Since KM 0/[M 0,M 0] is connected, (14.2) factors through a homomorphism >0,× (14.3) AM 0/[M 0,M 0] → R .

We denote the latter by 2ρQ0 . By a slight abuse of notation, we will denote by the same symbol the composite homomorphism 0 0 0 0 >0,× M (R) → M /[M ,M ](R) → AM 0/[M 0,M 0] → R and also the corresponding characters of Lie algebras. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 65

0 Explicitly, let us write n as a sum of aM 0/[M 0,M 0]-eigenspaces

0 M 0 n ' (n )α0 . α0 Then X 0 0 2ρQ0 = α · dim((n )α0 ). α0

14.1.2. Let us once and for all choose a trivialization of the line Λtop(g/q−). The key observation is the following: Proposition 14.1.3. There exists a canonically deﬁned G-invariant functional Z G : IQ0 (2ρQ0 ) → C. G/Q0

G Proof. By deﬁnition IQ0 (2ρQ0 ) is the space of scalar-valued functions on G that satisfy

−1 f(g · p) = (−2ρQ0 )(p ) · f(g). We claim that this vector space identiﬁes canonically with the space of top continuous degree diﬀerential forms on G/Q0. This follows from the fact that the action of Q0 on the top exterior 0 0 power of Te(G/Q ) ' g/q is given by 2ρQ0 . Now the sought-for functional is given by integration.

14.1.4. The homomorphism (14.3) has a well-deﬁned square root, which we denote by ρQ0 . We denote the normalized induction functor

n,G 0 0 IQ0 : Rep(M ) → Rep(G ) by n,G G IQ0 (W ) := IQ0 (W ⊗ (−ρQ0 )). n,G G Note that IQ0 (W ) and IQ0 (W ) are the same as K-representations. This is because −ρQ0 is trivial on KM 0 .

14.1.5. Let W be a representation of M and let W ∗ be its dual. From Proposition 14.1.3 we obtain that there exists a canonically deﬁned G-invariant pairing Z n,G n,G ∗ (14.4) IQ0 (W ) × IQ0 (W ) → C, f1, f2 7→ hf1, f2i. G/Q0

We clam: Proposition 14.1.6. Let W is admissible. Then the induced pairing

n,G K -ﬁn n,G ∗ K -ﬁn (IQ0 (W )) ⊗ (IQ0 (W )) → C

n,G ∗ K -ﬁn is perfect, i.e., identiﬁes (IQ0 (W )) with

n,G K -ﬁn ∗,alg ((IQ0 (W )) ) . 66 DENNIS GAITSGORY

Proof. It is enough to verify the statement at the level of K-representations. In the latter case, the pairing in question is given by IndK (L) ⊗ IndK (L∗,alg) → IndK ( ) → , KM KM KM C C where the latter map is the K-invariant (!) integration over K/KM ' G/P , or which is the same as unique splitting of the map ,→ IndK ( ) C KM C as K-representations. Now, for a given ρ ∈ Irrep(K), we have (IndK (L))ρ ' ρ ⊗ Hom (ρ, L) KM KM and similarly ∗ (IndK (L∗,alg))ρ ' ρ∗ ⊗ Hom (ρ∗, L∗,alg), KM KM and our pairing is the perfect pairing induced by ∗ ρ ⊗ ρ → C and ∗,alg HomKM (ρ, L) ⊗ HomKM (ρ, L ) → C. n,G 0 14.1.7. Let iQ denote the functor (m ,KM 0 )-mod → (g,K)-mod given by n,G G iQ0 (W ) := iQ0 (W ⊗ (−ρQ0 )). n,G G Note that iQ0 (W ) and iQ0 (W ) are the same as K-representations. Combining Proposition 14.1.6 with Theorem 14.3.6 (see below) and Proposition 11.1.2, we obtain:

0 Corollary 14.1.8. Let L be an admissible (m ,KM 0 )-module. Then there exists a canonical n,G ∗,alg n,G ∗,alg isomorphism between (iQ (L)) and iQ (L ). Remark 14.1.9. The assertion of Corollary 14.1.8 is purely algebraic. It amounts to the fact that the canonical K-invariant functional G 0 iQ0 (−2ρQ ) ' OK/KM → C is G-invariant. However, I was not able to give an algebraic proof of this fact: I needed to appeal to Proposition 14.1.6 that involves integration over the real manifold G/Q0. 14.1.10. Let us return to the situation of Sect. 14.1.5. Suppose that W is unitary. In this case, we have a G-invariant sesquilinear pairing Z n,G n,G (14.5) IQ0 (W ) × IQ0 (W ) → C, (f1, f2). G/Q0 n,G Interpreting the topologigal space underlying IQ0 (W ) as the space of continuous functions −1 f : K → W, f(k · m) = m · f(k), m ∈ KM , n,G we obtain that (14.5) equips IQ0 (W ) positive-deﬁnite scalar product, which is continuous in n,G the original topology on IQ0 (W ). NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 67

L2 n,G n,G Let IQ0 (W ) denote the completion of IQ0 (W ) with respect to the resulting L2-norm. L2 n,G We obtain that IQ0 (W ) is a unitary representation of G. Thus, normalized induction maps unitary representations to unitary representations.

14.2. Casselman’s submodule theorem.

14.2.1. We will prove the following: Theorem 14.2.2. Every irreducible (g,K)-module can be realized as a submodule of some G iQ(L) for an irreducible ﬁnite-dimensional (m,KM )-module L. By adjunction, this theorem is equivalent to:

Theorem 14.2.3. Let M be a finitely generated admissible (g,K)-module. Then the space Mn of n-coinvariants is non-zero. There exist two algebraic proofs of this theorem. One by O. Gabber and another by Beilinson- Bernstein via localization theory. We will present an analytic proof. It is based on considering the growth of matrix coefficients of K-finite vectors in a realization of our (g,K)-module.

14.2.4. Let a be a vector in a. Let V be a G-representation on a Banach space. Lemma 14.2.5. There exists a real number λ such that for all v ∈ V and v∗ ∈ V ∗ there exists a constant C so that we have ∗ ≥0 |hv , exp(t · a) · vi| ≤ C · exp(t · λ), t ∈ R .

Proof. Follows from the fact that the operator Texp(a) is bounded, and we take λ it be the log of its norm. 14.2.6. We shall say that a is regular if all of its eigenvalues on n are non-zero. We shall say that a is regular dominant if all of its eigenvalues on n are strictly positive. We have the following key assertion: Theorem 14.2.7. Let a ∈ a be regular. Let V be admissible, and let v and v∗ be K-ﬁnite such that hv∗, exp(t · a) · vi is not identically equal to zero. Then the set of λ ∈ R such that ∗ ≥0 |hv , exp(t · a) · vi| ≤ C · exp(t · λ), t ∈ R for some C is bounded below. In other words, this theorem says that the matrix coeﬃcient functions hv∗, exp(t · a) · vi cannot decay faster than all exponents.

14.2.8. Let us deduce Theorem 14.2.3 from Theorem 14.2.7. We will need the following assertion of independent interest, proved below. Proposition 14.2.9. Any admissible finitely generated (g,K)-module is finitely generated over U(n). Proof of Theorem 14.2.2. Pick a ∈ a to be regular anti-dominant. We pick a realization V of M. Fix v∗ ∈ (V ∗)K -fin. For v ∈ V K -fin such that hv∗, exp(t · a) · vi is not identically equal to zero, let λv be the infimum of those real numbers that |hv∗, exp(t · a) · vi| ≤ C · exp(t · λ) for some C. By Lemma 14.2.5 and Theorem 14.2.7, this infimum is well-defined. 68 DENNIS GAITSGORY

We claim that if v = u · v0 for u an element of U(n) with eigenvalue µ with respect to a (remember, it is negative), then

λv = λv0 + µ. Indeed, this is just the fact that exp(t · a) · v = exp(t · µ) · exp(t · a) · v0.

Hence, since M is ﬁnitely generated over U(n), the function

v 7→ λv attains a maximum. Let v0 ∈ M be a vector on which this maximum is attained. By the above, 0 it cannot be of the form u · v for u in the augmentation ideal of U(n). I.e., v0 projects to non-zero in Mn. Proof of Proposition 14.2.9. It is enough to show that any object of (g,K)-mod of the form U(g) ⊗ ρ, U(k) where ρ is a finite-dimensional representation of K, is finitely generated over U(n) ⊗ Z(g). I.e., it suffices to show that U(g) is finitely generated as a module over U(n) ⊗ U(k)op ⊗ Z(g). The latter is suffices to do at the associated graded level. I.e., it suffices to show that Sym(g) is finitely generated as a module over Sym(n) ⊗ Sym(k) ⊗ Sym(g)G. Since everything is positively graded, it suffices to show that Sym(g/n) is finitely generated over Sym(k) ⊗ Sym(g)G. However, the action of Sym(g)G on Sym(g/n) factors through Sym(a) (which is finite as a Z(g)-module), and the assertion follows. 14.3. Further realization results.

14.3.1. We will now consider the category (q,KM )-mod. It contains as a full subcategory (m,KM )-mod; it consists of those objects L on which n acts trivially. We have the pair of adjoint functors

(g,K) G Res :(g,K)-mod (q,KM )-mod : i . (q,KM ) Q

G The precomposition of iQ with (m,KM )-mod ,→ (q,KM )-mod is the functor that we earlier G denoted iQ. The proof of Proposition 12.2.3 applies and we have an isomorphism

Res(g,K) ◦ iG ' IndK ◦ Res(q,KM ). K Q KM KM

14.3.2. We have a similar situation at the level of group representations:

G G ResQ : Rep(G) Rep(P ): IQ .

As in Proposition 12.2.6, for a KM -admissible (q,KM )-module W , we have

G K -ﬁn G KM -ﬁn (IQ (W )) ' iQ(W ). NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 69

n -nilp 14.3.3. Inside the category (q,KM )-mod we single out the full subcategory (q,KM )-mod consisting of objects on which the Lie algebra n acts locally nilpotently. n -nilp Writing m as kM ⊕ a, we obtain that an object of (q,KM )-mod can be thought of as an algebraic representation of the group

ker(Q → M → TA), equipped with an action of a that commutes with KM and interacts with the N-action according to the adjoint action of a on N.

It is easy to see that any finite-dimensional object of (q,KM )-mod belongs in fact to n -nilp (q,KM )-mod . Indeed, the action of n is necessarily nilpotent because a has non-trivial adjoint eigenvalues on n. Moreover, it is easy to see that restriction defines an equivalence from the category of finite- dimensional representations of the (real Lie group) Q to that of finite-dimensional objects in (q,KM )-mod. 14.3.4. We will prove: Theorem 14.3.5. Let M be a (g,K)-module of finite length. Then there exists a finite- G dimensional object L ∈ (q,KM )-mod such that M embeds into iQ(L). As a consequence, as in we obtain (as in the case of Theorem 13.2.3): Theorem 14.3.6. And (g,K)-module of finite length can be realized as K-finite vectors in a Banach (or even Hilbert) representation of G. 14.4. Proof of Theorem 14.3.5. We will replace K by its connected component–this does not change the results of the preceding sections.

k 14.4.1. For an integer k, consider M/n · M as a (q,KM ∩ K0)-module. According to Proposi- tion 14.2.9, it is ﬁnite-dimensional. We will prove that there exists an integer k, such that the map G k M → iQ(M/n · M), (obtained by adjunction from the tautological map Res(g,K0) (M) → M/nk · M) is injective. (q,KM ∩K0) 14.4.2. Let n be a nilpotent Lie algebra. We consider the following functor on the category of n-modules: M 7→ Mb := lim M/nk · M. ← k By induction on the degree of nilpotence, from the Artin-Rees lemma we deduce: Lemma 14.4.3. The above functor is exact when restricted to the subcategory of ﬁnitely generated n-modules.

Let M be a (g,K0)-module of ﬁnite length. We claim:

Corollary 14.4.4. The map M 7→ Mb is injective. Proof. Lemma 14.4.3 and Proposition 14.2.9 reduce the assertion to the case when M is irreducible, which is what we will now assume. We now note that Mb also naturally acquires an g-action so that the natural map M → Mb is a map of g-modules (this is true for any g-module M). This follows from the fact that there 70 DENNIS GAITSGORY exists an integer k such that for any ξ ∈ g, if we take k times its bracket with elements from n, we get zero. For such k, the action of g is well-deﬁned as a map

0 0 M/nk +k · M → M/nk · M.

Recall also that if M is irreducible as a (h,K0)-module, then it is such as a g-module. Hence, it suﬃces to show that the map M → Mb is non-zero. However, we know that the composition M → Mb → M/n · M is non-zero. Indeed, this is equivalent to the statement of Theorem 14.2.2.

14.4.5. Consider now the map of (g,K0)-modules G k M → lim iQ(M/n · M). ← k It is injective because its composition with G k k lim iQ(M/n · M) → lim M/n · M = Mb ← ← k k is injective by Corollary 14.4.4. 0 G k Let Mk denote the kernel of the map M → iQ(M/n · M). We have 0 0 Mk+1 ⊂ Mk ⊂ ... Since M was assumed of ﬁnite length, this chain stabilizes. Hence its stable value M0 lies in G k 0 the kernel of all the maps M → iQ(M/n · M). But then M lies in the kernel of the map G k M → lim iQ(M/n · M). ← k Hence M0 = 0.

15. Week 9, Days 1 and (Tue., March 28 and Thurs., March 30) 15.1. Asymptotics of matrix coﬃcients.

15.1.1. Let V be an admissible representation, and v (resp., v∗) be a K-ﬁnite vector (resp., covector). We will be interested in the asympotic behavior of the function ∗ MCv∗,v(g) = hv , g · vi, where we will take g to be exp(a) for a regular anti-dominant element a ∈ a. We will prove:

Theorem 15.1.2. For every real number R there exists a ﬁnite collection of characters λn of a and polynomial functions pn on a such that the function

MCv∗,v(exp(a)) − Σ exp(λn(a)) · pn(a) n tends to zero faster than exp(R · min(α(a))). The characters λn appear as eigenvalues of a on α M/nk · M for a suﬃciently large integer k (here M = V K -ﬁn). The rest of this subsection is devoted to the proof of this theorem. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 71

15.1.3. Let us assume that V is a Banach representation of G. Since the operators Texp(a) are bounded, we can ﬁnd a real number R1 such that 0 ||Texp(a)|| < exp(−R · ρ(a)) for a ∈ a anti-dominant. Let m be such that m · min(α(a)) < ρ(a). Let the integer k be such that α k > R + m · R0. Consider the quotient M −→π M/nk · M. This is a ﬁnite-dimensional representation of the Lie algebra a; hence it integrates to a representation of the Lie group A.

k Let v1, ..., vn be vectors in M that project to a basis of M/n ·M. Then there exists polynomial functions pn on a such that the action of A on π(v) is given by

Σ exp(λn(a)) · pn(a) · π(vn), n

k where λn are among the eigenvalues of a on M/n · M. We will show that

∗ (15.1) MCv∗,v(exp(a)) − Σ exp(λn(a)) · pn(a) · hv , vni n decreases faster than exp(R · ρ(a)).

15.1.4. First, we claim: Lemma 15.1.5. The map i : M/nk ·M → V ∞/nk ·V ∞ intertwines the action of A on M/nk ·M (obtained by integrating the Lie algebra action) and the action of A on V ∞/nk · V ∞, coming from the action of A ⊂ G on V . Proof. For every v ∈ M/nk · M, the V ∞/nk · V ∞-valued functions on a, given by i(exp(a) · v) and exp(a) · i(v) take the same value at a = 0 and satisfy the same diﬀerential equation.

0 From this lemma, we obtain that the function (15.1) equals MCv∗,v0 (exp(a)), where v ∈ k ∞ 0 00 00 ∞ n · V . Write v as a sum of vectors of the form ξ1 · ... · ξk · v with ξi ∈ nαi and v ∈ V . We have

MCv∗,v0 (exp(a)) = Π exp(αi(a)) · MCv∗,v00 (exp(a)). i 0 Here MCv∗,v00 (exp(a)) increases slower than exp(−m · R · min(α(a))), while Π exp(αi(a)) de- α i creases faster that exp(k · min(α(a))). α

15.2. Growth conditions. 72 DENNIS GAITSGORY

15.2.1. Let λ be an element of a∗. We shall say that the exponents of an admissible (g,K)- module are ≥ λ if for every v, v∗, the characters λ0 that appear Theorem 15.1.2 satisfy: 0 Re(λ ) − λ ∈ Span ≥0 (positive roots). R The proof of Theorem 15.1.2 actually shows: Proposition 15.2.2. Let V be an admissible representation and let M be the (g,K)-module of its K-finite vectors. Then for a given λ ∈ a∗, the following conditions are equivalent: (i) The exponents of V are ≥ λ; (ii) For all the eigenvalues of a on M/n · M, we have 0 Re(λ ) − λ ∈ Span ≥0 (positive roots). R Here is a direct argument: Proof. Assume first that the condition on the eigenvalues of a on M/n · M is satisfied. Fix a regular anti-dominant a ∈ a∗. Let v ∈ M be such that the corresponding function

f(t) := MCv∗,v(exp(t · a)) has the asymptotic expansion

(15.2) f(t) = Σ exp(t · c) · pc(t) c with the maximal Re(c). It is easy to see that we can assume that v projects to a generalized a-eigenvector in M/n · M. To simplify the argument, we will assume that it projects to an eigenvector (in general, the idea is the same); let λ0 denote the corresponding eigenvalue. 0 Consider the function f (t). On the one hand, it equals MCv∗,a·v(exp(t · a)), and hence 0 0 λ (a) · MCv∗,v(exp(t · a)) + MCv∗,v0 (exp(t · a)), v ∈ n · M. In particular, the maximal real part of the exponent in the asymptotic expansion for MCv∗,v0 (exp(t · a)) is strictly < than that of f(t). Hence, f 0(t) − λ0(a) · f(t) has the maximal real part of the exponent in its asymptotic expansion strictly < than that of f(t). On the other hand, 0 0 0 0 f (t) − λ (a) · f(t) = Σ exp(t · c) · ((c − λ (a)) · pc − pc) c

If c0 denotes the c with the maximal real part, we obtain that 0 Re(c0) = λ (a), and hence

Re(c0) ≥ λ(a), as required. Vice versa, assume that the exponents are ≥ λ, and let λ0 be an eigenvalue of a on M/n · M. Let v ∈ M project to an eigenvector with the this eigenvalue. By the same logic as above, the maximal real part of the exponent in the asymptotic expansion of f 0(t) − λ0(a) · f(t) NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 73 is strictly < than that of f(t). And if c0 denotes the maximal part of the latter, we again obtain 0 that Re(c0) = λ (a), i.e., λ0(a) ≥ λ(a). Since this inequality is valid for any regular anti-dominant a, we obtain that 0 Re(λ ) − λ ∈ Span ≥0 (positive roots), R as required. Remark 15.2.3. Suppose that the exponents of V are ≤ λ for some λ. Note that this forces that the split part of ZG should act on V by a unitary character. Indeed, the condition implies that for every exponent, the real part of its restriction to zG is trivial. 15.3. Square-integrable representations.

15.3.1. An admissible (g,K)-module is said to be square-integrable if its exponents are ≥ (1 + ) · ρ for some ∈ R>0. 15.3.2. We have: Theorem 15.3.3. An admissible (g,K)-module is square-integrable if and only of all its matrix coeﬃcients lie in L2(G).

To simplify the exposition, we will prove this theorem in the case of G = SL2(R). Remark 15.3.4. The general case is not much more difficult; the main difference is that one needs to control the behavior of matrix coefficients near the walls of the anti-dominant cone; for that one proves (in essentially) the same way a version of Theorem 15.1.2 for an arbitrary parabolic (rather than the minimal parabolic). Proof. Consider the map K × exp(t · a, t > R) × K → G. This is an open embedding, the complement to whose image is compact. Let f(g) me a matrix coefficient of a (g,K)-module. We need to study the integrability of |f(g)|2 times the the pullback of the invariant top differential form under this map. Since f is K-finite on both sides, the question of integrability only depends on the further restriction of f(g) to t · a, t > R along the exponential map. Now, the required assertion follows from the next geometric assertion:

Lemma 15.3.5. The pullback of the invariant top differential form on G to R along the exponential map t 7→ exp(t · a) behaves like exp(−2t · ρ(a)) times a function that tends to 1 at t → ∞. Proof. We identify the tangent space to G at the point exp(t · a) with g via right translations. We need to calculate the determinant of the differential k ⊕ a ⊕ k → g at the point t · a. The differential in question acts as the embedding on the first two factors and the map

Adexp(−t·a) |k : k → g along the third factor. 74 DENNIS GAITSGORY

Consider another map k → g equal to the composition

Adexp(−t·a) k ,→ g n −→ n ,→ g, where the second arrow g n is the orthogonal projection. However, it is easy to see that the two maps differ by a map k → g that tends to 1 as t → ∞. Hence, it suffices to calculate the determinant of the map k ⊕ a ⊕ n → g, which equals to the identity on the first two factors and Adexp(−t·a) on n. However, the latter equals exp(−2t · ρ(a)), as required. 15.3.6. Let V be a square-integrable representation, and let M := V K -fin. Consider the matrix coefficient map M∗,alg ⊗ M → C∞(G).

By Theorem 15.3.3, its image consists of functions that belong to L2(G). Hence, we can introduce an inner form on M ⊗ M∗,alg by Z ∗ ∗ ∗ ∗ (15.3) (v1 ⊗ v1, v2 ⊗ v2) := hv1 , g · v1i · hv2 , g · v2i · µHaarG . G

We claim: Proposition 15.3.7. The above inner form on M∗,alg ⊗ M is (g ⊕ g,K × K)-invariant. Proof. The K × K-invariance follows immediately from the invariance of the Haar measure on G. To prove the (g ⊕ g)-invariance, we note that for, say a vector (0, ξ) ∈ g ⊕ g, we have: Z ∗ ∗ ∗ ∗ ∗ ∗ (v1 ⊗ ξ · v1, v2 ⊗ v2) + (v1 ⊗ v1, v2 ⊗ ξ · v2) = Lieξ(hv1 , g · v1i · hv2 , g · v2i) · µHaarG , G where Lieξ denotes the Lie derivative with respect to the corresponding left-invariant vector ﬁeld. We need show that the above integral vanishes. Note that this would be automatic if the support of the function ∗ ∗ f(g) = hv1 , g · v1i · hv2 , g · v2i was compact (which it never is unless G is compact). The issue is to show that f(g) decays fast enough so that the vanishing still holds. Since we only proved Theorem 15.3.3 in the case of SL2(R), we will be able to prove the assertion at hand only in this case. The assertion following from the next geometric claim: Lemma 15.3.8. Let f be a function on G that is K × K-ﬁnite, and assume that its restriction to R≥0 along t 7→ exp(t · a) (here a is an anti-dominant generator of a) decays faster than exp(−2(1 + ) · t) for some ∈ >0. Then R Z

Lieξ(f) · µHaarG = 0. G NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 75

Proof. We represent the integral µHaarG as the limit of integrals over closed subsets of G equal to the images of K × exp(0 ≤ t ≤ R) × K. For R > 0, this is a manifold with boundary, and we calculate the integral via the Stokes theorem. I.e., it equals the integral over the boundary, i.e., the subset K × exp(R) × K of the sub-top diﬀerential form

iξ(f · µHaarG ). This sub-top differential form is K × K-finite. So, it suffices to show that its value at the point exp(R) tends to 0 as t → ∞. However, this follows from the assumption on the decay of f. 15.3.9. Picking an arbitrary non-zero vector in v∗ ∈ M∗,alg, and restricting the above inner form to v∗ ⊗ M, we obtain that M acquires a (g,K)-invariant inner form. Applying Theorem 9.2.6, we obtain: Theorem 15.3.10. A square-integrable (g,K)-module can be realized as the space of K-finite vectors in a unitary representation. 15.4. Tempered representations. An admissible (g,K)-module is said to be tempered if it has exponents ≥ ρ. 15.4.1. We have: Theorem 15.4.2. The normalized induction of a tempered (g,K)-module is tempered.

We will only give a proof in the case of a split group G (e.g., G = SL2(R)) and the minimal parabolic (which in this case is the Borel subgroup). In this case, we start with a unitary chracater λ of T (see Remark 15.3.4) and we would like to see that the eigenvalues of the action of a ' t on n,G n,G rB ◦ iB (λ) have real part ≥ 0. We will prove a more precise result: n,G n,G Theorem 15.4.3. The eigenvalues of a ' t on rB ◦ iB (λ) are of the form w(λ) for w ∈ W . 15.4.4. To prove Theorem 15.4.3 we will have to revisit the Harish-Chandra homomorphism. Recall that the action of Z(g) acts on the module U(g) ⊗ U(t) U(b) via a homomorphism φ : Z(g) → Z(t) ' U(t) ' Sym(t). We now introduce the normalized Harish-Chandra homomorphism φn : Z(g) → U(t) by composing φ with the automorphism of U(t) ' Sym(t) induced by the map t 7→ t − ρ(t). I.e., for λ ∈ t∗, we have λ(φn(z)) = (λ − ρ)(φ(z)). We have: 76 DENNIS GAITSGORY

Theorem 15.4.5. The map φn maps Z(g) into the subalgebra of W -invariant functions in Sym(t). The resulting map Z(g) → Sym(t)W is an isomorphism. Proof. To show that the image of φn consists of W -invariant elements, it is enough to check this for every simple reﬂection si. Let P be a standard parabolic with Levi quotient M (in practice, we will take the sub-minimal parabolic corresponding to the simple root αi). Consider the corresponding Harish-Chandra homomorphism 0 φP : Z(g) → Z(m) ' Z(m ) ⊗ Sym(zM ). n We introduced the normalized version φP of φP by composing it with the automorphism of Sym(zM ), induced by the map

t 7→ t − ρP (t), where ρP is the character of zM equal to half the trace of its adjoint action on n(P ). It follows from the constuction, that n n n φ = φM ◦ φP , n where φM is the normalized Harish-Chandra homomorphism for the reductive group M. n This reduces the veriﬁcation of si-invariance of the image of φ to the case when G is of semi-simple rank 1. Splitting of the center, we can assume that G = SL2. In this case, the assertion is equivalent to the fact that for λ ∈ t∗, the action of Z(g) on the modules

λ λ M := U(g) ⊗ C U(b) and M −λ−2ρ is given by the same character. Moreover, since expression corresponds to the evaluation of a polynomial function at λ, we can assume that λ is a positive integer. However, in this case, the assertion follows from the fact that we have a non-zero map M −λ−2ρ → M λ; in fact we have a short exact sequence 0 → M −λ−2ρ → M λ → V λ → 0,

λ where V is the irreducible SL2-module of highest weight λ. Thus, φn is a homomorphism Z(g) → Sym(t)W , and it remains to show that it is an isomorphism. The latter is enough to do at the associated graded level. However, the maps gr(φ) and gr(φn) Sym(g)G → Sym(t)W agree, and both correspond to restricting G-invariant functions on g to W -invariant functions on t. The injectivity of this map is clear. The surjectivity follows from the fact that W -invariant functions on t are generated by characters of irreducible representations. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 77

15.4.6. Let us now prove Theorem 15.4.3: Proof. Let M be a (g,K)-module. By the construction of the normalized Harish-Chandra n,G homomorphism, the action of Z(g) on rB (M) (coming from the Z(g)-action on M) equals n n,G that obtained from φ and the Sym(t)-action on rB (M). n,G By adjunction, for an t-module N, the action of Z(g) on iB (N) equals that obtained from φn and the Sym(t)-action on N. Combining, we obtain that if Sym(t) acts on N by a character λ, then the eigenvalues λ0 of n,G n,G Sym(t) on rB ◦ iB (N) are such that they have the same projection as λ under the map Spec(Sym(t)) → Spec(Z(g)). However, this means that λ0 = w(λ) for w ∈ W . 16. Week 10, Day 1 (Tue, April 4) 16.1. Tempered vs square-integrable. 16.1.1. We now claim: Theorem 16.1.2. Any irreducible tempered (g,K)-module can be realized as a direct summand of a normalized induction of a square-integrable representation of the Levi quotient of a parabolic. Combining with Theorem 15.3.10 and Sect. 14.1.10, we obtain: Corollary 16.1.3. An irreducible tempered (g,K)-module can be realized as the space of K- finite vectors in a unitary representation. In addition, combining with Theorem 15.4.2 and Proposition 14.1.6, we obtain: Corollary 16.1.4. The dual of a tempered representation is tempered. 16.1.5. Proof of Theorem 16.1.2. To simplify the exposition, we will assume that G is split. Let M be an irreducible tempered (g,K)-module. It suffices to find a non-zero map n,G M → iP (N), where N is a square-integrable (m,KM )-module for some parabolic P with Levi quotient M. (Indeed, the induction is unitary, so any submodule splits off as a direct summand.) n,G Consider rQ (M) and consider the eigenvalues of t on it. For each such eigenvalue λ, write

Re(λ) = Σ ni · αi, where αi’s are positive simple roots and ni are non-negative.

Let λ0 be such that the number of i’s, for which the coeﬃcient ni is non-zero, is minimal. Let p ⊃ b be the parabolic subalgebra, where we add to b all the negative root spaces corresponding to i for which ni 6= 0 in the expansion of λ0. n,G Consider rP (M), and let Ne be its direct summand on which zM acts with generalized eigenvalues equal to λ0|ZM . Let N be some irreducible quotient of Ne. By construction, we have a surjective (in particular, non-zero) map n,G rP (M) → N, 78 DENNIS GAITSGORY

n,G and hence a map M → iQP (N). n,M It suffices to show that N is square-integrable. Let λ be an eigenvalue of t on rB∩M (N). By construction, Re(λ) = λ0 + Σ mi · αi = Σ (mi + ni) · αi, where αi’s are roots of M. Now, none of the mi + ni are zero, because it would contradict the minimality assumption on λ0. However, this by definition means that N is square-integrable. 16.2. The Langlands classification. Finally, here is the Langlands classification of all irreducible (g,K)-module in terms of the tempered ones. We will only state it in the case when G is split. 16.2.1. Let P be a standard parabolic of G with Levi quotient M. Let N be an irreducible tempered (m,KM )-module. Let λ be a real character of m/[m, m], which is regular anti-dominant. By definition, this means the following: Note that we have a surjective map t → m → m/[m, m]. We require that the evaluation of λ on every positive coroot of G, which is not a coroot of M, to be strictly negative. 16.2.2. The Langlands classification theorem says:

n,G λ Theorem 16.2.3. The induced (g,K)-module iP (N⊗C ) has a unique irreducible submodule. Every irreducible (g,K)-module arises in this way for a uniquely deﬁned triple (P, N, λ). The key point is the following assertion: Theorem 16.2.4. Let P − be the parabolic opposite to P and identify the Levi quotients of P and P − via P/N(P ) ' P ∩ P − ' P −/N(P −). n,G λ Consider the corresponding induced (g,K)-module iP − (N ⊗ C ). Then Hom(in,G(N ⊗ λ2 ), in,G(N ⊗ λ1 )) − 2 C P1 1 C P2 is non-zero only if P1 = P2, N1 ' N2 and λ1 = λ2. In the latter case, the above Hom is at most one-dimensional. Let us deduce Theorem 16.2.3 from Theorem 16.2.4. We will also need the following proposition, proved below: Proposition 16.2.5. For a (g,K)-module M there exists a parabolic P , an irreducible tempered (m,KM )-module N and a regular anti-dominant character λ of m/[m, m] such that M admits a n,G λ non-zero map to iP (N ⊗ C ). Proof of Theorem 16.2.3. First oﬀ, Proposition 16.2.5 implies that every irreducible (g,K)- n,G module M can be realized as a submodule in some i (N ⊗ λ1 ) for some triple (P , M , λ ). P1 1 C 1 1 1 Swapping the roles of P and P −, and using the duality property of normalized induction, i.e., Proposition 14.1.6 (and Corollary 16.1.4), we obtain that M can also be realized as a quotient n,G λ2 of some i − (N2 ⊗ C ) for some triple (P2, M2, λ2). P2 NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 79

Consider the composite map

in,G(N ⊗ λ2 ) M ,→ in,G(N ⊗ λ1 ). − 2 C P1 1 C P2

From Theorem 16.2.4, we obtain that P1 = P2, N1 ' N2 and λ1 = λ2. n,G 0 0 λ1 Suppose that M could be realized as a submodule of some potentially diﬀerent i 0 (N1⊗ ). P1 C Then we would still have a diagram

0 n,G λ2 n,G 0 λ1 i − (N2 ⊗ C ) M ,→ iP 0 (N1 ⊗ C ). P2 1 0 0 0 By Theorem 16.2.4, this implies P1 = P2 = P1, N1 ' N2 ' N1 and λ1 = λ2 = λ1. n,G λ Finally, suppose that a given iP (N ⊗ C ) has two diﬀerent irreducible submodules, i.e., we have an embedding n,G λ M1 ⊕ M2 ,→ iP (N ⊗ C ). Consider the corresponding surjections

n,G λ1 n,G λ2 i − (N1 ⊗ C ) M1 and i − (N2 ⊗ C ) M2. P1 P2 We obtain the non-zero maps

n,G λ1 n,G λ n,G λ1 n,G λ (16.1) i − (N1 ⊗ C ) → iP (N ⊗ C ) and i − (N2 ⊗ C ) → iP (N ⊗ C ). P1 P2

By Theorem 16.2.4, this implies that P1 = P = P2, N1 ' N ' N2 and λ1 = λ = λ2. Moreover, since in this case, the Hom in question is at most 1-dimensional, the maps (16.1) have the same image. However, the image of the ﬁrst was M1, and the image of the second was M2, a contradiction. Proof of Proposition 16.2.5. We will use the following combinatorial assertion (Langlands lemma): Lemma 16.2.6. Let λ be an element of (the real) t∗. Then there exists a unique subset I0 of the Dynkin diagram and a unique way to write λ as λ1 + λ2, where:

(i) λ1 is a character of m/[m, m] (where m is the Levi of the standard parabolic corresponding 0 0 to I − I ), and is regular anti-dominant as such. I.e., αˇi(λ1) < 0 for i ∈ I and αˇi(λ1) = 0 for i∈ / I0;

(ii) λ2 = Σ ni · αi, ni ≥ 0. i/∈I0 The proof of the lemma is obtained by applying the orthogonal projection from t∗ to the anti-dominant chamber, viewed as a convex subset of t∗. n,G Let M be a (g,K)-module. Consider rB (M), and consider the eigenvalues of t on it. Let λ be such that the corresponding Re(λ) is minimal in the order relation 0 00 0 00 λ ≥ λ ⇔ λ − λ ∈ Span ≥0 (α ). R i Consider the corresponding decomposition

Re(λ) = λ1 + λ2. Let P be the parabolic, whose simple roots correspond to the subset of the Dynkin diagram given by I − I0. 80 DENNIS GAITSGORY

n,G Consider rP (M), and let Ne be its direct summand corresponding to the eigenvalues of zM equal to the restriction of λ. Let N0 be some irreducible quotient of Ne. By construction, we have a non-zero map n,G 0 M → iP (N ). 0 By construction, the real part of the character of N with respect to zM is λ1, and so it is 0 −λ1 n,M 0 anti-dominant. N := N ⊗ C . The eigenvalues of t on rB∩M (N ) are of the form

λ + Σ mi · αi. i/∈I0

Moreover, Re(mi) ≥ 0 by the minimality assumption on λ. Hence, the eigenvalues of t on n,M 0 rB∩M (N ) are of the form λ2 + Σ mi · αi, Re(mi) ≥ 0. i/∈I0

n,M −λ1 This means rB∩M (N ⊗ C ) is tempered, as required.

Proof of Theorem 16.2.4. We will only give the proof when P1 and P2 are either all of G or the Borel subgroup (for the same reasons as in Theorem 15.4.2). By symmetry, it is enough to consider the following cases:

Case 1: P2 = G and P1 = B. In the case, by adjunction, the Hom in question equals

n,G µ1 λ1 Hom(rB (M2), C ⊗ C ), where µ1 is imaginary. n,G By assumption, the eigenvalues of t on rB (M2) have real part ≥ 0, whereas λ1 is strictly anti-dominant.

Case 2: P2 = B and P1 = B. By adjunction, the Hom in question equals

n,G n,G µ2 λ2 µ1 λ1 Hom(rB ◦ iB− (C ⊗ C ), C ⊗ C ), where µ1 and µ2 are imaginary.

n,G n,G µ2 λ2 By Theorem 15.4.3, the eigenvalues of t on rB ◦ rB− (C ⊗ C ) are of the form w(µ2 + λ2) for w ∈ W . Since both λ1 and λ2 are regular anti-dominant, w(µ2 + λ2) can equal µ1 + λ1 only for w = 1, µ2 = µ1 and λ1 = λ2 Thus, it remains to show that n,G µ λ n,G µ λ Hom(iB− (C ⊗ C ), iB (C ⊗ C )) is at most 1-dimensional. I don’t have a palatable argument at the moment... 16.3. Characters of representations. 16.3.1. Let V be an irreducible admissible representation of G on a Hilbert space. Recall the notion of operator of trace class, see Sect. 2.6.3. Our current goal is to prove the following theorem:

∞ Theorem 16.3.2. For f ∈ Cc (G), the corresponding operator Tf ∈ End(V ) is of trace class. The functional ∞ f 7→ Tr(Tf ,V ),Cc (G) → C is continuous, and only depends on the (g,K)-module V K -ﬁn. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 81

16.3.3. For the proof we will need to review a bit of theory. Let V be a Hilbert space. A continuous endomorphism S of V is said to be Hilbert-Schmidt if for some/any choice of orthonormal basis vi, the sum

2 2 (16.2) ||S||HS := Σ ||S(vi)|| i converges.

More generally, one deﬁnes the Hilbert-Schmidt inner form on the space of Hilbert-Schmidt operators by

(S1,S2)HS = Σ (S(vi),S(vi)), i 2 so that ||S||HS = (S, S)HS.

16.3.4. Here is a nice point of view of where this comes from. Consider the (algebraic) tensor product V ⊗ V ∗, and equip it with the inner form

∗ ∗ ∗ ∗ (v1 ⊗ v1 , v2 ⊗ v2 ) = (v1, v2) · (v1 , v2 ), where we note that V ∗ ' V via the inner form.

Deﬁne a map V ⊗ V → End(V ), v ⊗ v∗ 7→ (w 7→ v · hv∗, wi). Then this an isometric embedding

∗ V ⊗ V → End(V )HS, and one can show that the image is dense. I.e., the above embedding identiﬁes End(V )HS with the Hilbert space completion of V ⊗ V ∗.

16.3.5. We have the following assertion:

Proposition 16.3.6. (a) Any Hilbert-Schmid operator is compact.

(a’) Let vi be an orthonormal basis of V , and let S be and endomorphism of the algebraic span of the vi’s, so that the sum (16.2) converges. Then S continuously extends to a Hilbert-Schmid operator on V . 0 0 0 (b) For S ∈ End(V )HS and S bounded, we have S ◦ S ∈ End(V )HS and ||S ◦ S ||HS ≤ ||S||HS · ||S0||, and similarly for S0 ◦ S. (c) Any trace class operator is Hilbert-Schmidt.

(d) If S1,S2 are Hilbert-Schmidt, then S1 ◦ S2 is trace class and

† Tr(S1 ◦ S2) = (S1,S2)HS.

(e) For S is trace class and S0 bounded, then S ◦ S0 and S0 ◦ S are trace class and Tr(S ◦ S0) = Tr(S0 ◦ S). (e’) In the situation of (e), | Tr(S ◦ S0)| ≤ | Tr(S)| · ||S0||. 82 DENNIS GAITSGORY

16.3.7. Let us split V according to the K-isotypic components V ' ⊕ V ρ, ρ

ρ 2 and recall (see Theorem 10.4.5) that dim(V ) ≤ dim(ρ) . Let µρ denote the highest weight of ρ.

ρ Let Ck ∈ U(k) denote the Casimir operator of k. Note that Ck acts on V by the (positive) scalar λρ ≥ (µρ, µρ), where (−, −) is the inner form on k used to produce Ck. In particular, K -ﬁn 1 + TCk is an invertible endomorphism of V . The key trick in the proof of Theorem 16.3.2 is the following:

−n Proposition 16.3.8. For a sufficiently large integer n, the endomorphism (1 + TCk ) of V K -fin, extends continuously to all of V , and is Hilbert-Schmidt. Proof. As in Sect. 16.3.10 (see below), we may assume that the inner form on V is K-invariant. In particular, the different V ρ are pairwise orthogonal. By Proposition 16.3.6(a’), it suffices to find n so that the sum −2n ρ Σ (1 + λρ) · dim(V ) ρ converges.

m Recall also that dim(ρ) is bounded by ||µρ|| for m large enough. Excluding ﬁnitely many K-types, the above sum is dominated by

−2n+m Σ ||µρ|| . ρ This sum indeed converges, since it is a sum over the lattice in a positive quadrant in a Eucledian space. 16.3.9. Proof of Theorem 16.3.2. Note that as endomorphims of V ∞, we have:

∞ T n ◦ T n ◦ Tf ' T 2n , f ∈ C (G) (1+TCk ) (1+TCk ) l((1+TCk ) )(f) c where l(−) means the diﬀerential operator obtained from the given element of U(g) by left translations. Hence,

Tf = T −n ◦ T −n ◦ T 2n , (1+TCk ) (1+TCk ) l((1+TCk ) )(f) as operators on all of V .

Since T −n is Hilbert-Schmidt, we obtain that T −n ◦ T −n is trace class, (1+TCk ) (1+TCk ) (1+TCk ) by Proposition 16.3.6(d). Since T 2n is continuous, we obtain that Tf is trace class, l((1+TCk ) )(f) by Proposition 16.3.6(e). Note, furthermore, that the above argument goes through not only for smooth f, but for f 2n diﬀerentiable 4n times (need that l((1 + TCk ) )(f) ∈ Cc(G)). This shows the continuity of the map

f 7→ Tr(Tf ) in view of Proposition 16.3.6(e’). NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 83

K -ﬁn 16.3.10. To prove that Tr(Tf ) only depends on V , we argue as follows.

By averaging the given inner form on V with respect to K, we can assume that it is K- invariant. (It is easy to show that this change of inner form does not aﬀect the fact of Tf being of trace class V nor its trace).

Then

Tr(Tf ) = Σ ξρ ◦ Tr(Tf ) ◦ ξρ, ρ ρ where θρ is the projection on the V factor.

However, we saw in Corollary 11.1.5 that ξρ ◦ Tr(Tf ) ◦ ξρ only depends on the inﬁnitesimal equivalence class of ρ.

17. Week 10, Day 2 (Thurs., April 6) 17.1. Linear independence of characters.

17.1.1. We will prove the following theorem:

Theorem 17.1.2. Let Vα be a collection irreducible admissible representations, so that no two are inﬁnitesimally equivalent. Then their characters, viewed as distributions on G, are linearly independent.

The rest of this subsection is devoted to the proof of this theorem.

17.1.3. We argue by contradiction. Suppose there is a linear dependence. Replace the original set of α’s by a minimal subset such that the corresponding characters are still linearly dependent. ρ Let ρ be a K-type so that Vα 6= 0 for at least one α from the above subset. Consider the algebra

ξρ · H(G) · ξρ ⊂ HG, ρ where ξρ is the projector on V , see Sect. 11.1.3.

This algebra acts on V ρ for any admissible representation ρ. We claim:

Proposition 17.1.4. The map

ρ ξρ · H(G) · ξρ → ⊕ End (Vα ) α C is surjective.

Let us assume the proposition and ﬁnish the proof of the theorem. Let α0 be such that V ρ 6= 0. According to the above lemma, we can ﬁnd an element f ∈ ξ · H(G) · ξ such that it α0 ρ ρ acts as 1 in V ρ and as 0 in V ρ for α 6= α . α0 α 0

But then Tr(f, Vα0 ) 6= 0 and Tr(f, Vα) = 0, which is a contradiction. 84 DENNIS GAITSGORY

17.1.5. Proof of Proposition 17.1.4. As in Proposition 11.3.6, the functor M 7→ Mρ maps irreducible (pairwise distinct) (g,K)-modules with a non-trivial ρ-component to irreducible (pairwise distinct) ξρ · H(G) · ξρ-modules. ρ ρ In particular, for those of the α’s that Vα 6= 0, the ξρ · H(G) · ξρ-modules Vα are pairwise non-isomorphic. Now, the assertion follows from the next lemma:

Lemma 17.1.6. Let A be an associative algebra, and let Nα be a ﬁnite collection of irreducible ﬁnite-dimensional A-modules. Then the map

A → ⊕ End (Nα) α C is surjective. Proof. Follows from the fact that the only A-submodules in ∗ ⊕ Nα ⊗ Nα α are of the form ∗ ⊕ Nα ⊗ Mα, Mα ⊂ Nα, α ∗ and the identity element in Nα ⊗ Nα is not contained in any proper subspace of this form.

17.2. Regularity of characters.

17.2.1. Our next goal is the following fundamental result of Harish-Chandra’s: Theorem 17.2.2. The character of an irreducible admissible representation, viewed as a distribution on G, is given by integration against a locally L1-function; moreover, this function is C∞ on the regular semi-simple locus. 17.2.3. The actual theorem of Harsih-Chandra’s is even more precise than that. Note that the character Θ of a representation V is a distribution that is AdG-invariant. Furthermore, since V was assumed irreducible, by Schur’s lemma, the action of Z(g) on V ∞ is given by a homomorphism χ : Z(g) → C.

Hence, (17.1) z(Θ) = χ(z) · Θ, where in the left-hand side, we use the action of U(g) on Distr(G), given by the formula (u(d))(f) = d(l(u)(f)).

Harish-Chandra’s theorem says:

Theorem 17.2.4. Any distribution Θ on G which is AdG-invariant and satisﬁes (17.1) for some homomorphism χ : Z(g) → C is given by integration against a locally L1-function; moreover, this function is C∞ on the regular semi-simple locus. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 85

17.2.5. In the process of proof we will also see:

Theorem 17.2.6. The space of functions on the regular semi-simple locus that are AdG- invariant and satisfy (17.1) (for a given χ) is finite-dimensional. Combined with Theorem 17.1.2, this implies Theorem 10.2.6, i.e.: Theorem 17.2.7. The set of irreducible (g,K)-modules with a given character of Z(g) is finite. 17.3. More on the Harish-Chandra map. In this subsection we let G denote the complexification of our real reductive group. In order to avoid some silly technical difficulties, we will assume that the derived group of G is simply-connected (the initial G admits a finite cover by a group like this).

17.3.1. In order to disambiguate the notation, if λ is a character of t that integrates to a character of T , we will denote the latter by eλ. Let Greg-ss ⊂ G be the open subset consisting of regular semi-simple elements. Choose a Cartan subgroup T ⊂ G, and let T reg := T ∩ Greg-ss = {t ∈ T, α(t) 6= 1, ∀α}.

By the assumption that the derived group of G is simply-connected, eρ is a character of T . Let ∆ denote the function on T equal to eρ · Π (1 − e−α), α where the product is taken over the set of positive roots. Note that ∆ is W -equivariant against the sign character. (check for simple reflections, using the fact that αi is the only positive root that is flipped to negative by si). Note also that ∆ is invertible on T reg. Hence, −1 d 7→ Ad∆−1 (d) := ∆ · d · ∆ is a well-defined automorphism of Diff(T reg). The W -equivariance property of ∆ implies that this operation preserves W -invariant elements of Diff(T reg).

reg-ss 17.3.2. Note also that any AdG-invariant differential operator on G gives rise to a W - invariant differential operator on T reg. Indeed, the space of the latter identifies with the space of differential operators on T reg/W and restriction defines an isomorphism O(Greg-ss)G ' O(T reg)W .

Recall the normalized Harish-Chandra map φn : Z(g) → Sym(g). We are going to prove: Proposition 17.3.3. The following diagram commutes: Z(g) −−−−→ Diff(G)AdG −−−−→ Diff(Greg-ss)AdG   n  φ y y Sym(t)W Diff(T reg)W  x  Ad y  ∆−1 Diff(T )W −−−−→ Diff(T reg)W . 86 DENNIS GAITSGORY

Proof. Since everything in sight is affine, it is sufficient to check that the two differential operators on T reg resulting from the two circuits of the diagram from a given z ∈ Z(g) act in the same way on W -invariant functions (defined on all of T ). We take our supply of functions to be given by characters of irreducible fin.dim. G-representations. Thus, f be the character of the irreducible representation with highest weight λ. On the one hand, by the definition of φn, we have z(f) = hλ + ρ, φn(z)i · f. On the other hand, the Weyl character formula says −1 w(λ+ρ) f|T reg = ∆ · Σ sign(w) · e . w∈W Hence, for u ∈ Sym(t)W , and the corresponding element d ∈ Diff(T reg)W , we have −1 w(λ+ρ) (Ad∆−1 (d))(f|T reg ) = ∆ · hλ + ρ, ui · Σ sign(w) · e = hλ + ρ, ui · f|T reg , w∈W as desired. 17.4. Digression: conjugacy classes of tori and regular semi-simple elements.

17.4.1. We return to the case when G is a real reductive group. We know that all Cartan subgroups in GC are conjugate. But this is not the case for G.

For example, for G = SL2(R), there is the split Cartan (diagonal matrices), and the compact Cartan (which equals the maximal compact). Here is the general picture.

17.4.2. Consider the aﬃne variety

AdG G// AdG := Spec(O(G) ). Let reg (G// AdG) ⊂ G// AdG be the regular locus. Namely, under the complexiﬁcation W (G// AdG)C ' TC//W := Spec(O(TC) ), reg (G// AdG) corresponds to C T reg/W ' T reg//W ⊂ T //W, C C C where the ﬁrst isomorphism is due to the fact that the action of W on T reg is free. C reg Note that (G// AdG) is typically disconnected. For example, for G = SL2(R),

G// AdG ' R via g 7→ Tr(g), and reg (G// AdG) = R − {2, −2}. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 87

17.4.3. The preimages of points under the (tautological) map reg-ss reg G → (G// AdG) reg-ss are called stable conjugacy classes. In other words, g1, g2 ∈ G belong to the same conjugacy class if and only if there exists an element g ∈ GC such that Adg(g1) = g2. However, stably conjugate does not mean conjugate. For example, for an element k in the −1 maximal compact of SL2(R), it is stable conjugate to k , but not conjugate. Lemma 17.4.4. Every stable conjugacy class consists of ﬁnitely many conjugacy classes. Proof. Let g be a regular semi-simple element, and let T be its centralizer; this is a compact subgroup of G. Then the set of conjugacy classes in the stable conjugacy class of t is the quotient of the set of

{g ∈ GC, σ(g) · g ∈ TC} by the action of TC g, t 7→ g · t and the action of G g, g1 7→ g1 · g. Hence, the above quotient injects into 1 H (Z/2Z,TC), where Z/2Z acts on TC by σ. However, the latter cohomology group is ﬁnite. Indeed, writing T as 0 → T 1 → T → T 2 → 0, with T 1 a split form and T 2 a compact form, we have H1( /2 ,T 1) = 0 (by Hilbert 90), where Z Z C as H1( /2 ,T 2) ' ±1 ⊗ Λ, Z Z C Z where Λ is such that T 2 = Λ ⊗ . C Gm Z 17.4.5. For a regular semi-simple element g ∈ G, its centralizer is a Cartan subgroup in G. One can show: Proposition 17.4.6. Elements of the same stable conjugacy class give rise to conjugate Cartan subroups. Proof. Let g be a regular semi-simple element and let T be its stabilizer. Then T (viewed as a real torus), and T 0 be the split part of T . Replacing G by the centralizer of T 0, we can assume that T is compact modulo the center. In the latter case it is known that T is the only conjugacy class of Cartan subgroups that are compact modulo the center. 17.4.7. Thus, we obtain a map reg (G// AdG) → {Conjugacy classes of Cartan subgroups}. However, it is easy to see that this map factors via a map reg π0((G// AdG) ) → {Conjugacy classes of Cartan subgroups}, reg i.e., points in the same connected component of (G// AdG) give rise to conjugate Cartans. In particular, we obtain that the set of conjugacy classes of Cartan subgroups in G is ﬁnite. 88 DENNIS GAITSGORY

17.4.8. Let be a conjugacy class of Cartan subgroups; fix a representative T. Let reg-ss reg-ss G ⊂ G be the subset consisting of those elements that can be conjugated into T. reg-ss reg-ss Then G is open and closed in G (i.e., it is the union of some of the connected reg-ss components). Indeed, G equals the preimage under reg-ss reg G → (G// AdG) of reg reg (G// AdG) ⊂ (G// AdG) , the latter being the union of those connected components that give rise to . reg-ss The quotient G / AdG is defined as a real analytic manifold, and we have a finite covering map reg-ss reg (17.2) G / AdG → (G// AdG) whose fibers are conjugacy classes within a given stable conjugacy class.

17.4.9. Here is what this picture looks like for SL2(R). As we remarked above, there are two conjugacy classes of Cartans: the split one and the compact one. The subset reg-ss reg-ss Gsplit ⊂ G is that of hyperbolic elements (i.e., diagonalizable matrices with distinct eigenvalues). It consists of two connected components (the eigenvalues being positive or negative). Equivalently, they are the elements g with Tr(g) > 2 and Tr(g) < −2, respectively. The corresponding map (17.2) is an isomorphism. The subset reg-ss reg-ss Gcomp ⊂ G is that of elliptic elements. It is connected. It consists of elements g satisfying −2 ≤ Tr(g) ≤ 2. The corresponding map (17.2) is a 2-fold covering. 17.5. Smoothness of the distribution on the regular semi-simple locus. In this subsection we will prove the easy part of Theorem 17.2.4, namely that for any distribution Θ on G which is AdG-invariant and satisﬁes (17.1) for some homomorphism χ : Z(g) → C, its restriction to Greg-ss is given by an analytic function. We will also prove Theorem 17.2.6.

17.5.1. Since Θ is AdG-invariant we can regard it as a distribution on real analytic manifold reg-ss G / AdG. Since the action of Z(g) commutes with the AdG-action, we have a well-defined map reg-ss Z(g) → Diff(G / AdG), and out distrubition saitsfies (17.3) z(Θ) = χ(z) · Θ, z ∈ Z(g), reg-ss viewed as distributions on G / AdG. The key claim is the following: NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 89

reg-ss reg-ss Proposition 17.5.2. Locally on G / AdG, the ring Diff(G / AdG) is finitely generated reg-ss over O(G / AdG) and Z(g). 17.5.3. Let us deduce the smoothness of Θ (and the finite-dimensionality of such Θ’s) from Proposition 17.5.2. reg-ss Working locally on G / AdG (i.e., over some open), choose a set of generators d0, d1, ..., dk reg-ss reg-ss of Diff(G / AdG) as a module over O(G / AdG) and the image of Z(g) with d0 = 1.

Then for a basis of vector ﬁelds ξ1, ..., ξn, we have l l ξi · dj = Σ Σ fj0 · dj0 · zj0 , j0=0,...,k l 0 l l where l runs over some ﬁnite set depending on i, j, j , and fj0 is a function and zk ∈ Z(g).

Consider the system of linear ﬁrst order diﬀerential equations on a tuple (Θ0, ..., Θk) l (17.4) ξi(Θj) = Σ Σ fj0 · χ(zj0 ) · Θj0 . j0=0,...,k l

Then any distributional solution of this system is smooth. And the dimension of the space of solutions is bounded by k + 1. Now, if Θ is a distributional solution of (17.3), then

{Θi := di · Θ} is a solution of (17.4).

In particular, Θ = Θ0 is smooth, and the dimension of the space of such is bounded by k +1.

18. Week 11, Day 1 (Tue, April 11) 18.1. Proof of Proposition 17.5.2. 18.1.1. Pulling back with respect to reg-ss reg (18.1) G / AdG → (G// AdG) , it is easy to see that the statement of the proposition for the original homomorphism reg-ss Z(g) → Diﬀ(G / AdG) is equivalent to one obtained by composing with d 7→ ∆−1 · d · ∆.

Note that since (17.2) again is a covering map, we can pull back differential operators from reg reg-ss (G// AdG) to G . Now, it follows from Proposition 17.3.3 that the above homomorphism reg-ss Z(g) → Diff(G / AdG) (after conjugation by ∆) is obtained in this way from a homomorphism reg (18.2) Z(g) → Diff((G// AdG) ), which is a real version of n φ W reg reg Z(g ) ' Sym(t ) → Diff(T //W ) → Diff(T //W ) ' Diff((G// AdG) ). C C C C C 90 DENNIS GAITSGORY

Since (18.1) is a covering map, its suﬃces to prove the corresponding ﬁnite generation statement for (18.2).

reg reg 18.1.2. For each , the subset Diff((G// AdG) ) receives a finite covering map from T . reg Hence, it is enough to prove the finite generation assertion on T . Since Sym(t) is finitely generated over Sym(t)W , it suffices to prove our assertion for the latter replaced by the former.

However, it is obvious that Sym(t) and O(T) generate Diﬀ(T).

18.2. The local L1-property: an introduction. We now wish to prove the more diﬃcult part of Theorem 17.2.4, namely that Θ is given by integration against a locally L1 function.

This will result from the combination of the following three assertions.

18.2.1. First, we recall that since |∆| is well-deﬁned as a function on G// AdG, we can pull it back to G and obtain an AdG-invariant function.

We will prove:

Theorem 18.2.2. The function |Θ · ∆|, which is a C∞ function on Greg-ss is locally bounded on G, i.e., it restriction to U ∩ Greg-ss is bounded, whenever U ⊂ G is compact.

Theorem 18.2.3. The inverse of |∆| is locally an L1 function, i.e., the integral Z −1 f 7→ f · |∆ |, f ∈ Cc(G) Greg-ss is absolutely convergent and deﬁnes a continuous functional Cc(G) → C.

Theorem 18.2.4. If a distribution on G which is AdG-invariant and satisﬁes (17.1) vanishes when restricted to Greg-ss, then it is zero.

18.2.5. Let us see how these statements combined yield the desired local L1 property of Θ.

Proof. By combining Theorems 18.2.2 and 18.2.3, we obtain that Z Θ(e f) := f · Θ Greg-ss is absolutely convergent and deﬁnes a continuous functional Cc(G) → C.

It remains to see that Θ(f) = Θ(e f). However, Θ − Θ0 is a distribution that vanishes on Greg-ss. Hence, it is zero by Theorem 18.2.4.

18.3. Digression: the Grothendieck-Springer resolution and the Steinberg variety. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 91

18.3.1. Let G be a complex reductive group. Let X denote its flag variety, thought of as the variety of Borel subgroups in G. For x ∈ X we let Bx ⊂ G denote the corresponding Borel subgroup. Recall that the Grothendieck-Springer resolution, denoted Ge, is the the closed subvariety in G × X consisting of pairs (g, x) such that g ∈ Bx. We have the tautological maps q p G ←− Ge −→ X. The map q is proper. Its fiber over a given g ∈ G is the variety of Borel subgroups that contain g; equivalently, this is the fixed-point set Xg of g ∈ G acting on X. Over Greg-ss, the map is an étaleGalois cover with Galois group W . The map p is a smooth fibration. In fact that it is what is called the tautological fibration: the fiber over x ∈ X is Bx. We have a natural projection r from Ge to the abstract Cartan T .

18.3.2. The Grothendieck-Steinberg variety StG is deﬁned to be Ge × G.e G In other words, this is a closed subvariety in G × X × X consisting triples (g, x1, x2) such that g ∈ Bx1 ∩ Bx2 . w For w ∈ W , let StG be the locally closed subvariety of StG corresponding to the condition that the Borels x1 and x2 are in relative position w. The map w w StG → (X × X) is a smooth ﬁbration, where (X × X)w ⊂ (X × X) is the corresponding G-orbit (Schubert cell), i.e., the variety of pairs (x1, x2) in relative position w. w w w Every point of StG is contained in exactly one StG. Let StG denote the closure of StG in w w StG. The subvarieties StG are the irreducible components of StG.

18.3.3. Let now G be a real group. We consider the following hybrid, denoted ReStG. This is the closed subvariety in

GC × XC equal to

GeC ∩ G × C.

Equivalently, if σ is the involution on GC corresponding to G, then ReStG is the subset of

GC × XC × XC equal to the intersection of StG , G × X × X and the subvariety given by the condition that C C C σ(x1) = x2. w Let ReStG denote the intersection w ReStG ∩ StG . C w w Let ReStG denote the closure of ReStG. 92 DENNIS GAITSGORY

18.3.4. Suppose that for a given w ∈ W , the variety Xw := {x, σ(x) ∼w x} C w is non-empty (which is a necessary condition for ReStG to be non-empty). One can show that Xw is the disjoint union of (ﬁnitely many) G-orbits. C

In particular, we obtain that G has ﬁnitely many orbits on XC. ω For a G-orbit ω on X , let Xω be its preimage in ReSt , and let ReSt denotes its closure. C C G G

18.3.5. To an orbit ω one can attach a particular real form Tω of TC. Namely, choose a point x ∈ Xω, and consider the group C (18.3) Bx ∩ Bσ(x).

It is solvable, and its canonical toric quotient identiﬁes with TC. Now, (18.3) has a canonical real structure, given by swapping the factors. This real structure induces one on the toric quotient. A diﬀerent point x0 on the same orbit will give rise to the same real structure: if g · x = x0, then the action of g isomorphes the two situations. 18.3.6. It follows from the construction that for a given ω, the map

r : GeC → TC restricts to a map ω ReStG → Tω, and hence also to a map ω ReStG → Tω. ω Furthermore, if (g, x) ∈ ReStG is such that g is regular semi-simple, then the centralizer T of g on G is contained in Bx, and the the resulting map

T ⊂ Bx ∩ G → Tω is an isomorphism.

ω 18.3.7. Vice versa, given a point x ∈ X , the intersection Bx ∩ Bσ(x) is a subgroup of GC compatible with its real structure and containing a Cartan subgroup. Hence,

Bx ∩ Bσ(x) ∩ G contains a Cartan subgroup of G. This Cartan projects isomorphically to Tω. 18.4. Proof of Theorem 18.2.2. 18.4.1. Let reg-ss ReStG ⊂ ReStG reg-ss be the preimage of G ⊂ G under the map q : ReStG → G. The map q is proper and surjective. Hence, it suﬃces to see that the pullback of |Θ · ∆|, reg-ss viewed now as a continuous function on ReStG , is locally bounded. For that, it is suﬃcient to show that for for every ω, the pullback of |Θ · ∆| to ω reg-ss ReStG ∩ ReStG ω extends to a continuous function on ReStG. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 93

ω reg-ss 18.4.2. First, we claim that the restriction |Θ·∆| to ReStG ∩ ReStG is the pullback of a func- reg ω reg-ss tion Θe ω on Tω . In fact, this is the case for any AdG-invariant function on ReStG ∩ ReStG :

ω reg-ss reg Indeed, any two points in ReStG ∩ ReStG that map to the same point in Tω are uniquely conjugate by a unipotent element of GC, which then necessarily belongs to G.

Thus, it suﬃces to show that Θe ω extends continuously to all of Tω.

18.4.3. We claim that the function Θe ω satisﬁes the diﬀerential equation

n (18.4) φ (z)(Θe ω) = χ(z) · Θe ω, z ∈ Z(g).

This follows via Proposition 17.3.3 from the fact that Θ| w reg-ss itself satisﬁes the ReStG ∩ ReStG diﬀerential equation z(Θ)e = χ(z) · Θe, z ∈ Z(g).

reg 18.4.4. Finally, we claim that any function on Tω that satisﬁes (18.4) extends continuously (in fact, as a smooth function) to all of Tω.

Indeed, this follows as in Sect. 17.5.3: such a function can be realized as one coordinate of a solution of a vector-valued system of 1st order linear diﬀerential equations with smooth coeﬃcients.

−1 18.5. Proof of the local L1 property of |∆| .

18.5.1. We need to show that for every compact subset U ⊂ G, the integral of |∆|−1| over U ∩ Greg-ss (with respect to the Haar measure on G) is convergent. Let Ue denote the preimage of U under the map q : ReStG → G.

Since the map reg-ss reg-ss q reg-ss ReStG := ReStG ×G → G G is a ﬁnite covering map, it is enough to prove the convergence of the corresponding integral over reg-ss Ue ∩ ReStG , with respect to the pullback of the Haar measure (we can pullback measures along local isomorphisms).

reg-ss 18.5.2. Note that ReStG (unlike all of ReStG) is smooth, and the above measure on it is given by a top diﬀerential form, equal to the pullback of the bi-G-invariant top diﬀerential form on G.

Note that the top diﬀerential form on G is obtained by restriction from an (algebraic) top diﬀerential form ω on G . Let ω denote the pullback of ω along G C Ge G

q : GeC → GC. reg-ss Then our top diﬀerential form on ReStG is obtained by restriction along

ReStreg-ss ,→ Greg-ss ,→ G . G eC eC 94 DENNIS GAITSGORY

18.5.3. Note that ∆ is a well-defined (algebraic) function on TC. Its locus of zeros is the divisor T − T reg. C C Note that we have a Cartesian diagram Greg-ss −−−−→ G eC eC     y y T reg −−−−→ T . C C In particular, the complement to Greg-ss in G is cut out by (the pullback) of ∆. eC eC Note that the map q is ramified over G − Greg-ss. Hence, ω has a zero along the divisor C C Ge G − Greg-ss. Hence, we obtain that eC eC ω0 := ω · ∆−1 Ge Ge is a well-defined algebraic differential form on GeC.

18.5.4. Hence, out task is to show that for any compact Ue ⊂ ReStG, the integral Z |ω0 | Ge reg-ss Ue∩ReStG is convergent. However, this follows from the next general assertion: Lemma 18.5.5. Let Y be a real algebraic variety of dimension k. Let ω be an algebraic k- diﬀerential form on Y . Then for any compact subset U ⊂ Y (R), the integral Z |ω|

sm U∩Y (R) is convergent. The proof can be obtained from the Noether normalization lemma.

18.6. Proof of the uniqueness of extension (for SL2(R)). Our current task is to prove Theorem 18.2.4. We will only do this in the case of G = SL2(R). 18.6.1. We ﬁrst consider the open subset Greg = G − {±1}. It contains as an open Greg-ss, and the complement Greg − Greg-ss is the union of two G-orbits: one is the orbit of the non-trivial unipotent element, and the other its translate by −1. reg We will ﬁrst show that if Θ is an AdG-invariant distribution on G , supported on the closed subset Greg − Greg-ss, and satisfying z(Θ) = χ(z) · Θ, then Θ = 0. We will consider the case when Θ is supported on the regular unipotent orbit. The other case is similar. NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 95

reg 18.6.2. The key observation is that G / AdG is still a well-defined differentialble manifold. reg The distribution Θ descend to a well-defined distribution, denoted Θe on G / AdG. reg The regular unipotent orbit corresponds to a point x ∈ G / AdG; let t denote a local coordinate around this point so that t(x) = 0. Being a distribution supported at one point, Θe has the form k (18.5) Σ ak · ∂t · δx. k≥0

reg An element z ∈ Z(g) gives rise to a diﬀerential operator on G / AdG. Take z = Cg, the reg Casimir element. The corresponding diﬀerential operator on G / AdG is of order 2; write it as 2 2 d := (b0 · ∂t + b1 · t · ∂t ) + (c0 · ∂t) + terms with higher order of vanishing with respect to t.

Lemma 18.6.3. We have: b0 = 0, b1 = 2, c0 = 3. Let us assume the lemma and ﬁnish the proof of the theorem.

18.6.4. Let K be the largest integer such that aK 6= 0 in (18.6). When we apply to it an operator 2 (b1 · t · ∂t ) + (c0 · ∂t) + terms with higher order of vanishing with respect to t we get a distribution 0 k (18.6) Σ ak · ∂t · δx, K+1≥k≥0 where 0 aK+1 = aK · (c0 − (K + 1) · b1). In particular, (18.6) cannot be eigen-distribution for the above operator unless

c0 = (K + 1) · b1.

However, in our case c0 = 3 and b1 = 2 and no such K exists, since 2 is not divisible by 3.

18.6.5. Proof of Lemma 18.6.3. We will determine the coeﬃcients by acting by Cg on some particular AdG-invariant functions. Namely, let f be such a function with the ﬁrst order of vanishing at x. We need to show: d(f 2) (18.7) (d(f 2))(x) = 0, (d(f)((x) = 3, (x) = 10. f We take f(g) = fe(g) − 2, fe(g) = Tr(g). Recall that 1 C = E · F + F · E + · H2. g 2 Hence, 3 2 2 Cg(fe) = · f,e Cg(fe − 1) = 4 · (fe − 1). 2 This implies (18.7). 96 DENNIS GAITSGORY

18.6.6. Finally, we are left to deal with the case that Θ is a distribution supported at ±1 ⊂ G. We claim that no such distribution cab be eigen for Cg. We will deal with the case of 1 ∈ G; the case of −1 is similar. The space of distributions supported at 1 is isomorphic to U(g), and the action of Z(g) corresponds to multiplication. Now the assertion follows from the fact that U(g) has no zero divisors.