REPRESENTATION THEORY of REAL GROUPS Contents 1

Home , Borel subgroup, Real form (Lie theory)

ZIJIAN YAO

Contents 1. Introduction1 2. Discrete series representations6 3. Geometric realizations9 4. Limits of discrete series 14 Appendix A. Various cohomology theories 15 References 18

1. Introduction 1.1. Introduction. The goal of these notes is to survey some basic concepts and constructions in representation theory of real groups, in other words, “at the infinite places”. In particular, some of the definitions reviewed here should be helpful towards the formulation of the Langlands program. To be more precise, we first fix some notation for the introduction and the rest of the survey. Let G be a connected reductive group over R and G = G(R). Let K ⊂ G be an R-subgroup such that K := K(R) ⊂ G is a maximal compact subgroup. Let G0 ⊂ GC be the 1 unique compact real form containing K. By a representation of G we mean a complete locally convex topological C-vector space V equipped with a continuous action G × V → V . We denote a representation by (π, V ) with π : G → GL(V ). 2 Following foundations laid out by Harish-Chandra, we focus our attention on irreducible admissible representations of G (Definition 1.2), which are more general than (irreducible) unitary representations and much easier to classify, essentially because they are “algebraic”. More precisely, by restricting to the K-finite subspaces V fin we arrive at (g,K)- modules. By a theorem of Harish-Chandra (recalled in Subsection 1.2), there is a natural bijection between Irreducible admissible G- Irreducible admissible representations (g,K)-modules ←→∼ infinitesimal equivalence isomorphism

In particular, this allows us to associate an infinitesimal character χπ to each irreducible admissible representation π (Subsection 1.3), which carries useful information for us later in the survey. The strategy to understand unitary representations (up to infinitesimal equivalence) is therefore clear: first classify all irreducible admissible (g,K)-modules, then distinguish unitary ones

1 For example, take G = GLn, K = O(n) and G0 = U(n). The corresponding Cartan involutions of the split real form G and compact real form G0 are respectively the trivial one and the Chevalley involution, up to GLn(C)-conjugation. 2In the majority of the notes we will pretend that (π, V ) is a Hilbert space representation for simplicity. 1 2 from the rest. The first step is more or less fully understood by work of Harish-Chandra, Lang- lands and others. In particular, Langlands proves the “Langlands classification”, as well as the archimedean Langlands correspondence, which states that there is a natural bijection between L-packets of irreducible admissible (Relevant) Langlands pa- representations of G rameters of G ←→∼ infinitesimal equivalence conjugation by G∨

For quasi-split groups G over R, every Langlands parameter is relevant. For G = GLn, each L-packet consists of a single (infinitesimal equivalence class of) admissible representations. The second step, namely to classify all unitary representations, seems more difficult and has not been completed. One of the simplest classes of irreducible admissible representations are discrete series – they unitary representations that are square integrable in a suitable sense. Such representations form a discrete subset in the unitary dual Gb of G (under the Fell topology), hence the name. They are in a certain sense infinite dimensional analogs of highest weight representations. We discuss discrete series in Section2, where examples of SL 2 / GL2 are considered Using the theory of global characters, Harish-Chandra showed that discrete series exist precisely when rank(G) = rank(K). Moreover, one can describe discrete series representations of G via the Harich-Chandra parameters, as stated in Theorem 2.9. We then discuss explicit realizations of discrete representations. The classical story of realizing highest weight representations of compact groups are given by the Borel–Weil–Bott’s theorem, which states that if G is a compact real Lie group with flag variety X = GC/B for some complex Borel subgroup B ⊂ GC, then the coherent cohomology of holomorphic line bundles Lλ is non-trivial precisely when λ is regular after a ρ-shift, in which case is appears in one degree and is the irreducible representation of highest weight λ (for the precise statement and its proof see Theorem 3.1). Moreover, all highest weight representations of G appear this way. When G is non-compact, Langlands conjectured that a similar story should hold when X is replaced by the open subdomain D which is an orbit under the G-action 3 and when the coherent cohomology is replaced by the L2-cohomology. Langlands predicted that the corresponding irreducible representation of G should be a discrete series representation and this procedure exhausts all discrete series. Langlands’s conjecture is now a theorem of Schmid, which we explain in Subsection 3.2. Finally, we end this survey by a brief discussion on limits of discrete series and some properties of (non-generate) limits of discrete series. Acknowledgement. Most of the results stated in the survey are due to Harish-Chandra and Schmid. Due to the depth of the theorems stated in the survey, only brief outlines are provided instead of detailed proofs. I would like to thank professor Schmid for patiently explaining many of his results to me, which helps me immensely in understanding his proofs and organizing this survey article.

1.2. Admissible representations. Let G be a reductive group over R as in the introduction, let G = G(R) and g its Lie algebra. 1.2.1. Smooth vectors. Let V = (V, π) be a representation of G. Recall that a vector v ∈ V is 1 d C if for each X ∈ g, its derivative Xv = dt exp(tX)v t=0 exists. This allows us to inductively deﬁne Ck. The smooth vectors in V form a subspace V sm = {v ∈ V v is Ck for all k ≥ 1} ⊂ V.

3D = D(∆+) depends on a choice of positive roots, or equivalent a Borel subalgebra b ⊂ g 3

In other words, these are vectors v such that g 7→ π(g)v defines a smooth map G → V . It is not difficult to see that V sm is a subrepresentation of G. Moreover, the representation of g on V sm by taking derivatives is a Lie algebra representation. ∞ Next recall the action of the Hecke algebra H = Cc (G) on V given by Z π(f)v := f(g)π(g)v dg G for any f ∈ H and v ∈ V . First we observe that this “averaging” process makes the vectors sufficiently smooth. More precisely, we have π(f)v ∈ V sm for each f ∈ H, v ∈ V . To see this, rewrite d Z Xπ(f)v = f(g) π exp(tX)g v dg dt G t=0 d Z = f exp(−tX)g π(g)v dg dt G t=0 Z = fX (g) π(g)v dg = π(fX )v G d ∞ sm where fX (g) := f(exp(−tX)g) , which lies in Cc (G). Therefore π(f)v ∈ V by induction. dt t=0 From this it allows that Corollary 1.1. The subrepresentation V sm ⊂ V is dense. 1.2.2. K-finite vectors. Restricting a representation to its compact subgroups is a valuable tool in representation theory, and in our setup leads to the notion of (g,K)-modules. First let us recall that a representation (V, π) of G is unitary if it is a Hilbert space representation such that the action of G is unitary. Let V = (V, π, h · i) be a Hilbert space representation, then V |K is unitarizable: we define a Hermitian form ( · ) by Z (v1, v2) := hπ(g)v1, π(g)v2idg, K which is K-invariant by construction. It induces the same topology on V as the original Her- mitian form h · i. Let σ be an equivalent class of irreducible representation of K, then the σ-isotypic component of V is ∼ V (σ) = {v ∈ V : Chπ(K)vi = σ}. A vector v ∈ V is K-finite if it is contained in V (σ) for some irreducible representation σ of K. The subspace of K-finite vectors in V is V fin := ⊕ V (σ) σ∈Kb where the (algebraic) direct sum is taken over equivalent classes of irreducible representations of K 4. Definition 1.2. A representation (π, V ) of G is admissible if

• π|K is unitary; • each V (σ) appearing in V fin has finite multiplicity/dimension. Remark 1.3. From the discussion above, for a Hilbert space representation, to check admissi- bility it suffices to check the second condition.

4By our discussion on unitarizability above, the equivalent classes of irreducible representations can be identiﬁed with the unitary dual Kb 4

Theorem 1.4 (Harish-Chandra). Suppose that G is connected reductive as above, then every irreducible unitary representation of G is admissible. 1.2.3. (g,K)-modules. The K-finite vectors V fin ⊂ V in an admissible representation (V, π) (which is usually much smaller) is a lot easier to analyze. Note that the G-action does not in general preserve K-finiteness, so we consider its Lie algebra action and are led to the following definition Definition 1.5. A(g,K)-module is a vector space V with a g (as Lie algebra) action and K action satisfying the following conditions

(1) V is a countable algebraic direct sum V = ⊕iVi with each Vi a ﬁnite dimensional K- invariant sub-representation. 5 (2) For any X ∈ k = Lie(K) and v ∈ V , we have d X · v = π exp(tX)v dt t=0 (3) For any k ∈ K,X ∈ g and v ∈ V , we have π(k) X · π(k−1)v = (Ad(k)X) · v

We similarly define admissible (g,K)-modules – namely V is admissible if the Vi’s in part (1) of the definition can be chosen to be distinct K-representation. We first claim (as a sanity check) that for an admissible Hilbert representation, its K-finite vectors V fin indeed produces a (g,K)-module Lemma 1.6. Let (V, π) be a Hilbert space representation of G. Then V fin ∩ V sm ⊂ V is dense, and is preserved by the action of g. If (V, π) is admissible, then V fin ⊂ V sm. An admissible representation is irreducible if and only if its associated (g,K)-module is irreducible. It is a result of Harish–Chandra that each irreducible admissible (g,K)-module arises as the K-finite subspace of an admissible G-representation. Definition 1.7. Two admissible representations are infinitesimally equivalent if their associated (g,K)-modules are isomorphic. This completes the explanation of the first bijection diagram in the introduction, which we repeat below Corollary 1.8. Irreducible admissible G- Irreducible admissible representations (g,K)-modules ←→∼ infinitesimal equivalence isomorphism We also have a corresponding bijection for unitary representations. Theorem 1.9 (Harish-Chandra). Two irreducible unitary representations are infinitesimally equivalent if and only if they unitarily equivalent. Therefore, when restricting to unitary representations, we get Irreducible unitary G- Irreducible unitary admissible representations (g,K)-modules ←→∼ unitary equivalence isomorphism

5This is equivalent to the condition that for each v ∈ V, Kv spans a ﬁnite dimensional continuous subrepresentation in V . 5

1.3. Inﬁnitesimal characters.

1.3.1. Universal enveloping algebras. Let gC be a Lie algebra over C, its universal enveloping U(gC) can be constructed as the quotient of the tensor algebra T (gC) by ideals generated by I = {XY − YX − [X,Y ] x, y ∈ g}. U(gC) is not graded but inherits the ﬁltration from T (gC). ∗ By the Poincare–Birkhoﬀ–Witt theorem, the graded morphism T (gC) → Gr (U(gC)) factors through the symmetric algebra on gC

T (gC) Sym(gC)

∗ Gr (U(gC)) and induces an isomorphism of (commutative) algebras ∼ ∗ Sym(gC) −→ Gr (U(gC)).

This in particular implies that gC ,→ U(gC) is injective. + Corollary 1.10. Let hC ⊂ gC be a Cartan subalgebra, and fix a choice ∆ ⊂ ∆ of positive roots. Then there is a direct sum decomposition + − U(gC) = U(hC) ⊕ (U(gC)n + n U(gC)) + − where n = ⊕α∈∆+ gα denotes the positive root spaces, and likewise for n . 1.3.2. The Harish-Chandra isomorphism. The Harish-Chandra isomorphism describes the center Z = Z(U(gC)) of the universal enveloping algebra. First we define the algebra homomorphism 0 ∼ γ : Z → U(gC) −→ U(hC) −→ Sym(hC) where the middle map is the projection (Corollary 1.10). Harish-Chandra observed that, by introducing the following twist, one obtains a symmetric description of Z. The idea is to define

hC −→ U(hC) by X 7−→ X − ρ(X) 1 P where ρ = 2 α∈∆+ α is the half-sum of the positive roots. Note that ρ(X) is a number. This extends to an automorphism of commutative algebras h : U(hC) → U(hC). Definition 1.11. The Harish-Chandra homomorphism is 0 γ := h ◦ γ : Z −→ Sym(hC). Twisting by ρ has the effect that for any z ∈ Z, γ(z) is invariant under the action of Weyl group W . This morphism γ is in fact canonical (in particular independent of choices of ∆+). Moreover, we have Lemma 1.12 (Harish-Chandra). The Harish-Chandra homomorphism induces an isomorphism ∼ W γ : Z = Z(U(gC)) −→ (Sym(hC)) . 1.3.3. Infinitesimal characters. Now consider any λ ∈ h∗ , which gives rise to an algebra homo- C morphism λ : U(hC) → C. Composing λ with the Harish-Chandra homomorphism we get γ λ χλ := λ ◦ γ : Z = Z(gC) −→ U(hC) −→ C, which we call the infinitesimal character of weight λ. ∗ Lemma 1.13. Every algebra homomorphism χ : Z → C is of the form χ = χλ for some λ ∈ h . 0 C Moreover, χλ = χλ0 if and only if λ = wλ for a Weyl group element w ∈ W . 6

Now suppose we have an irreducible admissible representation π of G, then the center Z = Z(gC) acts on V by scalars (following from a form of Schur’s lemma), therefore we get a homomorphism χπ : Z → C. By the lemma above, χπ = χλ for some λ. We call χπ (or λ, by abusing terminology) the inﬁnitesimal character of π.

2. Discrete series representations Throughout this section we assume that (π, V ) is an irreducible admissible Hilbert space representation of G.

2.1. Discrete series. Now we turn to discrete series representations. We keep our notation of G ⊂ GC as a real form and g ⊂ gC. Recall that, a matrix coefficient for a given Hilbert representation (π, V ) is a function of the form mv1,v2 : G → C given by g 7→ hπ(g)v1, v2i for v1, v2 ∈ V . Lemma 2.1 (Godement). Let (π, V ) be an irreducible unitary representation of G, then the following are equivalent 2 (1) The matrix coefficients mv1,v2 lies in L (G) (namely is square integrable) for every v1, v2 ∈ V ; 2 (2) mv1,v2 lies in L (G) for some nonzero v1, v2 ∈ V ; (3) There exists a G-equivariant embedding V,→ L2(G) where L2(G) carries the right regular representation of G. Definition 2.2. An irreducible unitary representation is a discrete series if it satisfies the equivalent conditions above.

In the example of representations of GL2(R) listed in Appendix ??, the representations Dk are discrete series when k ≥ 2. 6

2.2. Harish-Chandra (or global) characters. Let (π, V ) be an irreducible admissible rep- ∞ resentation as before, then for each f ∈ Cc (G), the “Hecke” operator π(f) ∈ End(V ) given by Z π(f)v = f(g)π(g)v dg G is bounded, and in fact a trace class operator. This allows us to deﬁne

Θπ(f) = Tr π(f) ∞ as a linear functional Θπ : Cc (G) → C. The distribution Θπ is the Harish-Chandra character (or global character) of π. Before we explain the relevance of Harish-Chandra characters in the study (in particular the parametrization) of discrete series, we note the following “linear independence” of characters Theorem 2.3 (Harish-Chandra). Two irreducible admissible representations of G are inﬁnites- imally equivalent if and only if they have the same Harish-Chandra characters.

The universal enveloping algebra U(gC) acts on Θπ via diﬀerentiation of the distribution.

Lemma 2.4. Z(gC) acts on Θπ via the inﬁnitesimal character of π. More precisely, for all X ∈ Z(gC), XΘπ = χπ(X)Θπ.

6 D1 is what we call a limit of discrete series. 7

∞ A distribution θ : Cc (G) → C which is invariant under the conjugation action of G and such that Z(gC) acts by scalars is called an invariant eigendistribution. By conjugation invariance of trace, Θπ is an invariant distribution under the action G, therefore it is an invariant eigendistribution. A crucial property of such distributions, proved by Harish-Chandra, is the regularity theorem which states that Theorem 2.5 (Harish-Chandra). Any invariant eigendistribution Θ on G is locally integrable. In other words, Θ is given by Z Θ(f) = f(g)θ(g)dg, G where θ is locally integrable. Moreover, θ is real analytic on the set of regular elements reg G := {g ∈ G : dim ker(Ad(g) − Ig) = rank(G)}. Another property of invariant eigendistribution is that they are completely determined by restrictions to (the regular elements in) the Cartan subgroups of G. By conjugation invariance, we only need to look at one Cartan subgroup in each conjugacy class. 2.3. Formula for Harish-Chandra characters. In this Subsection we will use explicit forms (and other properties) of the characters Θπ to determine an “upper bound” for discrete series. For simplicity let us assume that (after choosing ∆+), the half sum of positive roots ρ := 1 P 7 2 α∈∆+ α lies in the weight lattice. As usual let W be the Weyl group. Let H ⊂ G be a Cartan subgroup. Then after the choice of positive roots ∆+ ⊂ ∆, we define a function Y Y D = “ (eα/2 − e−α/2)” := eρ 1 − e−α α∈∆+ α∈∆+ on H, called the Weyl denominator (relative to ∆+). Following the discussion from the previous subsection, let Θπ be the global character of an irreducible admissible representation (π, V ) with infinitesimal character λ (or any irreducible invariant eigendistribution Θ satisfying XΘ = χλ(X)Θ). Let τ be the function given by τ(h) = θ(h)D(h) on Hreg. Then on the subset G He reg := gHregg−1 = {ghg−1|h ∈ Hreg, g ∈ G}, g∈G/H we have the expression for the analytic function θ θ(ghg−1) = τ(h)/D(h) for all g ∈ G, h ∈ Hreg with D(h) 6= 0. Harish-Chandra shows that 8 τ is a linear combination of exponentials with constant coefficients 9 on Hreg. We state this more precisely as Lemma 2.6 (Harish-Chandra). Retain notations from above, then τ is of the following form on Hreg: X wλ τ = τH = awe w∈W where aw are constants. If moreover H is compact, then τ extends to an analytic function on all of H.

7 This is not a serious assumption. If GC is simply connected then it is automatic. In general, a suitable 2-cover of GC will also satisfy this. We will not bother with the general situation, though it is not difficult. 8 by considering the differential equation γ(X)τ = χλ(X)τ where γ : Z → U(hC) is the Harish-Chandra homomorphism 9more generally polynomial coefficients, for non-irreducible invariant eigendistributions 8

This result is an ingredient of the “only if” direction in the following Theorem 2.7 (Harish-Chandra’s criterion). The group G has discrete series representations if and only if rank(G) = rank(K), in other words, G has a compact Cartan subgroup. Remark 2.8 (Outline of the “only if” direction). For this we need the following notion: an invariant eigendistribution Θ decays at ∞ if for each H ⊂ G, θ · |D| tends to 0 away from compact subsets of H. The direction of interest follows from the following two claims

(1) If π is a discrete series representation, Θπ decays at ∞. (2) If an invariant eigendistribution Θ decays at ∞ then there exists a compact Cartan T such that θ|T 6= 0. One way to prove the first claim is to show that Θπ extends to a bounded operator on the Sobolev space Sn(G) for n sufficiently large, and use the Weyl integration formula. Claim (2) follows from the lemma above: if H is non-compact, and Θπ decays at infinity, then the coefficients of τ are forced to be 0. Example. We list some examples that are most relevant for automorphic representations. GK rank (G) rank (K) n SLn(R) SO(n) n − 1 b 2 c p+q p q SO(p, q) S(O(p) × O(q)) b 2 c b 2 c + b 2 c Spn(R) U(n) n n SU(p, q) S(U(p) × U(q)) p + q − 1 p + q − 1 In particular, • SLn(R) has discrete series if and only if n = 2 (see Appendix ??); • SO(p, q) has discrete series if and only if at least one of p, q is even; • Spn(R) and SU(p, q) always admit discrete series representations. 2.4. Harish-Chandra parameters. We want to “classify” all discrete series representations. For this it suffices to assume that G contains a compact Cartan T ⊂ K ⊂ G, note that K is uniquely determined (instead of up to conjugacy) once T is fixed. Their corresponding Lie algebras (and complexifications) are denoted by t, k, g (respectively tC, kC, gC). + Let ∆ be the root system for (gC, tC), and fix a choice of ∆ ⊂ ∆. Let ∆c be the compact roots (corresponding to (kC, tC)) and ∆n = ∆\∆c the non-compact roots. This defines also 1 P 1 P λ ρ = + α and ρ = + α = ρ−ρ . Under the exponential map λ 7→ e , we identify c 2 α∈∆c n 2 α∈∆n c the weight lattice Λ ⊂ it∗ ⊂ t∗ with the character group X∗(T ), which is the same as algebraic C characters of TC. Suppose that π is a discrete series whose character Θπ does not vanish on T . Now we consider the restriction of Θπ to this compact subgroup T , which in fact determines Θπ by a theorem of Harish-Chandra (and in turn determines π up to infinitesimal equivalence). Suppose that 10 χπ = χλ+ρ , then we have P w(λ+ρ) awe Y θ| = w∈W ,D = (eα/2 − e−α/2). T D α∈∆+

Since θ is Wc invariant (D is Wc-skew), we can deduce (from the linear independence of global characters) that there are at most #|W/Wc| discrete series with inﬁnitesimal character χλ+ρ. We can now state Harish-Chandra’s parametrization of discrete series, which says that, for each λ + ρ that is regular, there are precisely #|W/Wc| discrete series, and these account for all discrete series representations.

10note the ρ shift compared to the convention before 9

Theorem 2.9 (Harish-Chandra). Suppose that G contains a compact Cartan T . Suppose λ ∈ Λ is such that λ + ρ is regular. Then there exists a unique invariant eigendistribution Θλ+ρ which decays at ∞ and has the following explicit form over T : P ε(w)ew(λ+ρ) θ | = (−1)q w∈Wc . λ+ρ T Q α/2 −α/2 hα,λ+ρi>0 e − e

Every such Θλ+ρ with λ + ρ regular is a discrete series character (with inﬁnitesimal character λ + ρ.) Conversely, every discrete series character is one of these Θλ+ρ with λ + ρ regular. In other words, we have a bijection {Discrete Series} (X∗(T ) + ρ)reg ←→∼ inf. equiv Wc ∗ Deﬁnition 2.10. An element µ ∈ X (T ) + ρ /Wc is called a Harish-Chandra parameter. ∗ reg A Harish-Chandra parameter µ is regular if it lies in X (T )+ρ /Wc, otherwise it is irregular. For a regular Harish-Chandra parameter µ = λ+ρ, we denote by π(µ) = π(λ+ρ) the discrete series representation corresponding to λ + ρ under the bijection above. In the next Section we will try to construct the backward arrow by geometric realization of each π(λ + ρ), following the work of Schmid.

3. Geometric realizations 3.1. The Borel–Weil–Bott theorem. Let us recall the classical story of geometric realization of irreducible representations of compact connected semisimple Lie groups GR (namely the irreducible holomorphic representations of GC). We adopt the following notation for this section • GC is a connected complex Lie group with gC = Lie(GC). + • Fix a choice of positive roots ∆ ⊂ ∆, denote by b = hC ⊕n the Borel sub-algebra where P α hC is the Cartan and n = α∈∆− g • Let B (resp. HC) be the corresponding Borel (resp. Cartan) subgroup of GC. In 11 particular B = {g ∈ GC : Ad(g)b = b}. Let X be the ﬂag variety of GC (whose complex points parametrize Borel sub-algebras). The ∼ choice of b above makes X a homogeneous space of GC, inducing the identiﬁcation X = GC/B. We identify the weight lattice Λ ⊂ h∗ with X∗(H ) via the exponential map as in 2.4. For C C C × each λ ∈ ΛC, we get a character of the Borel λ : B → HC → C . This allows us to construct a GC-equivariant holomorphic line bundle

Lλ −→ X over X, where

Lλ = GC × C/(gb, z) ∼ (g, λ(b)z). Note that, for example, L−2ρ is the canonical bundle on X. Let O(Lλ) be the sheaf (of holomorphic sections) of Lλ. Theorem 3.1 (Borel–Weil–Bott). ∗ H (X, O(Lλ)) 6= 0 ⇔ λ + ρ is regular In which case, the cohomology is non-vanishing in degree p = #(λ + ρ) := #{α ∈ ∆+ : hλ + ρ, αi < 0},

11The notation may seem confusing – B, b, n are complex Lie groups/algebras. We do not consider their real forms in this subsection. 10

p moreover, the representation GC on H (X, O(Lλ)) is precisely the highest weight representation with weight w(λ + ρ) − ρ, where w ∈ W is the element of the Weyl group so that w(λ + ρ) is dominant. Sketch of Proof. First one proves a special case (namely the Borel–Weil theorem): for dominant 0 λ, H (X, O(Lλ)) is an irreducible representation of GC of highest weight λ. Let GR be a compact form of GC, then HR = HC ∩ GR is a maximal torus. By compactness ∼ ∼ of GR, we know that GR acts transitively on X, and we have X = GC/B = GR/HR (here B is the complex parabolic subgroup stabilizing b). Therefore, L =∼ G × where H acts λ R HR Cλ R Cλ λ × by HR ,→ HC −→ C . It is clear that

∞ ∼ ∞ HR C (X, Lλ) = (C (GR) ⊗ Cλ) , so the question is essentially how to describe the holomorphic sections among the C∞ ones (namely asking for Cauchy–Riemann equations in this setup). It turns out the holomorphic P α sections are precisely ones killed by the action of n = α∈∆− g , in other words, we have a GR equivariant isomorphism

0 ∞ HR H (X, O(Lλ)) = {f = f ⊗ 1 ∈ (C (GR) ⊗ Cλ) : r(n)f = 0} where r(n) denotes the inﬁnitesimal right translation of X ∈ n. By compactness of GR and the Peter–Weyl theorem, we know that C∞(G ) ⊂ ⊕ V ⊗ V ∗, but H0(X, O(L )) is ﬁnite R b i∈Gb i i λ dimensional, so R 0 M ∗ H (X, O(Lλ)) ⊂ Vi ⊗ Vi .

i∈GbR One deduces that n o 0 M ∗ HR H (X, O(Lλ)) ⊂ Vi ⊗ ψ ∈ Vi ⊗ Cλ : nψ = 0 . i The special case (Borel–Weil theorem) therefore follows from the fact that nψ = 0 picks out the lowest weight space (by choice of n). Now to go from the special case to the computation of all cohomology groups, we proceed by induction. + α Fix a simple root α ∈ ∆ and define pα := b ⊕ g , which is a parabolic subalgebra of gC. Let P := N (p ) be the corresponding parabolic subgroup and let X = G /P . Now the α GC α α C α inclusion B ⊂ Pα induces a GC-equivariant fiber bundle ∼ π ∼ GC/B = X −→ Xα = GC/Pα, 1 whose fiber is isomorphic to P (C) which is the flag variety for sl2(C). The case of Borel–Weil– Bott theorem for SL2 can be directly verified. Now we take the Leray spectral sequence p,q q p p+q E2 = H (Xα, OXα (Vλ)) ⇒ H (X < O(Lλ)) p where Vλ is the holomorphic vector bundle given by p H (Xy, OX (Lλ )) y Xy −1 at each point y, where Xy = π (y). Using the theorem for SL2 (via direct computation) we p,∗ know that E2 is non-vanishing for at most one p, so the spectral sequence collapses and it follows that k ∼ M q p H (X, O(Lλ)) = H (Xα, OXα (Vλ)). p+q=k Now consider sα ∈ W the reflection in α, and the decomposition of k H (X, O(Lsα(λ+ρ)−ρ)) 11 as above. By comparing the two decompositions, and use the fact that •V 0 =∼ V1 , λ sα(λ+ρ)−ρ •V 1 = mV 0 = 0, λ sα(λ+ρ)−ρ we obtain that, if hλ + ρ, αi > 0 then k ∼ k+1 H (X, O(Lλ)) = H (X, O(Lsα(λ+ρ)−ρ)). Both sides are 0 if hλ + ρ, αi = 0. This implies the theorem.

3.2. Geometric realizations of discrete series. Keep our notation of T ⊂ K ⊂ G ⊂ GC as + in Subsection 2.4. Fix ∆ ⊂ ∆, which determines b = tC ⊕ n, with n the negative (complex!) ∼ root space. In the case when G is non-compact, G no longer acts transitively on X = GC/B. However it is still true that D = D(∆+) := G-orbit of b is open in X. In fact we have D =∼ G/T and the embedding ∼ ∼ D = G/T ,→ X = GC/B is the Borel embedding. For different choices of positive roots, D(∆+) = D(∆e +) if and only if + + ∆ and ∆e are conjugate under the action of the compact WEyl group Wc. For an example, 1 consider the case of SL2, where we have H ⊂ P (C). For λ ∈ Λ ⊂ it as in the previous sections, we define the holomorphic line bundle Lλ as before + and restrict it to D = D(∆ ). By abusing notation we still denote Lλ = Lλ|D. Therefore we have a G-equivariant holomorphic line bundle + Lλ → D(∆ ). 2 The non-compactness of D naturally leads us to consider the L cohomology of Lλ. Note that there is a natural map from L2 cohomology to coherent cohomology ∗ ∗ H(2)(D, O(Lλ)) −→ H (D, O(Lλ)), which is in general neither injective or surjective. + Theorem 3.2 (Schmid, Langlands’s conjecture). Let D = D(∆ ) and Lλ be defined as above, then ∗ H(2)(D, O(Lλ)) 6= 0 ⇔ λ + ρ is regular In which case, the cohomology is non-vanishing in one degree 12 + + p(λ + ρ) := #{α ∈ ∆c : hλ + ρ, αi < 0} + #{α ∈ ∆n : hλ + ρ, αi > 0}. p In this case, the representation G on H(2)(D, O(Lλ)) is a discrete series representation with Harish-Chandra character Θλ+ρ. In view of Theorem 2.9, we have p(λ+ρ) ∼ H(2) (D, O(Lλ)) = π(λ + ρ). Remark 3.3. If G is compact then D = X and L2 cohomology agrees with coherent cohomology ∗ ∼ ∗ H(2)(X, O(Lλ)) −→ H (X, O(Lλ)), so Schmid’s theorem recovers Theorem 3.1. We first very briefly outline the proof of Schmid’s theorem.

12This number p(λ + ρ) agrees with #(λ + ρ) in the case when G is compact. It is in fact the number of negative eigenvalues of a certain curvature form of Lλ+ρ. 12

fin sm Lemma 3.4. For each Vi ∈ Gb in the unitary dual and λ ∈ Λ, the inclusion Vi ,→ Vi (see Subsection 1.2) induces an isomorphism T T ∗ fin ∼ ∗ sm H (n,Vi ) ⊗ Cλ −→ H (n,Vi ) ⊗ Cλ between (finite dimensional) Lie cohomologies.

Idea of proof. From Cartan decomposition gC = kC ⊕ pC we get two spectral sequences p,q,fin p q fin p+q fin E2 = H (n ∩ pC,H (n ∩ kC,Vi )) ⇒ H (n,Vi ) p,q,sm p q sm p+q sm E2 = H (n ∩ pC,H (n ∩ kC,Vi )) ⇒ H (n,Vi ) fin sm and a functorial morphism on the E2 page, induced by Vi ,→ Vi . One needs to show that on the E2 level this induces an isomorphism (after applying T -invariance with Cλ-twisted coefficients), for details we refer to [8]. Note that the finite-dimensionality is clear from this consideration. Recall one formulation of Plancherel’s theorem (its compact incarnation, the Peter–Weyl theorem, was used in the proof of the theorem of Borel–Weil–Bott), Z 2 ∼ ∗ L (G) = Vi⊗bVi dµ(i). i∈Gb Lemma 3.5. There is an G-equivariant isomorphism Z T ∗ ∼ p ∗,sm H(2)(D, O(Lλ)) −→ Vi⊗b H (n,Vi ) ⊗ Cλ dµ(i). i∈Gb Moreover, the isomorphism can be made isometric after choosing suitable inner product on the right hand side. Idea of proof. The hermitian metric on D gives rise to an inner product on the antiholomorphic cotangent bundle which we identify with n. Now we consider the coboundary operator ^ ∗,fin ^ ∗,fin di∗ : Hom( n,Vi ) −→ Hom( n,Vi ). ∗ ∗ The inner product on Vi and n allows us to define the formal adjoint (di∗ ) of di∗ . Let i∗ = ∗ ∗ p ∗ di∗ (di∗ ) + (di∗ ) di∗ and its kernel be H (n,Vi ) := ker i∗ , then we claim that Z T ∗ ∼ p ∗ H(2)(D, O(Lλ)) = Vi⊗b H (n,Vi ⊗ Cλ dµ(i) i∈Gb Z T ∼ p ∗,sm = Vi⊗b H (n,Vi ) ⊗ Cλ dµ(i). i∈Gb For the second isomorphism, we need to show that

p ∗ T ∼ p ∗,sm T H (n,Vi ⊗ Cλ −→ H (n,Vi ) ⊗ Cλ , the proof of this is slightly involved and is streamlined after the proof of Hodge theorem, replacing 2 ∞ ∗ ∗,sm the role of L (S) (resp. C (S)) for a compact complex manifold S by Vi (resp. by Vi ). For details we refer to [8] again.

Lemma 3.6. For each Vi ∈ Gb, let Θi be the Harish-Chandra character associated to Vi and θi the corresponding function on T . Then we have Y α X p p ﬁn 1 − e θi = char T (−1) H (n,Vi ) . α∈∆+ 13

Idea of proof. The proof uses a spectral sequence argument as in the proof of Lemma 3.4. Using the finite-dimensionality of the E2 terms, we may replace θi by K-characters over T , namely the formal sum of the global characters of the K-irreducible constituents of Vi. The lemma then follows from the compact case, by applying a theorem of Harish-Chandra (from [2]), which reg roughly states that the K-character and θi agree on K ∩ G . We also need the following result p ∗,sm T Lemma 3.7 (Casselman–Osborne). If H (n,Vi ) ⊗ Cλ 6= 0 for some p then Vi has infinitesimal character χλ+ρ. Lemma 3.5 and Lemma 3.7 together imply the following p Corollary 3.8. H(2)(D, O(Lλ)) is a finite direct sum of discrete series representations, all with infinitesimal character χλ+ρ. Proof. This follows from the following fact (due to Harish-Chandra [1]): an irreducible unitary representation (V, π) is discrete series if and only if the plancherel measure assigns the singleton {π} a strictly positive measure in the decomposition of L2(G). Now by Lemma 3.7, the integral of the right hand side of Lemma 3.5 is a finite direct sum, so it consists only of discrete series representations. It remains to determine when this is non-zero, the multiplicity of each irreducible component, and their Harish-Chandra characters. ∗ Lemma 3.9. If λ + ρ is singular, then H(2)(D, O(Lλ)) = 0.

Idea of proof. The point is to consider the (virtual) multiplicity of Vi in the alternating sum P p p 2 p(−1) H(2)(D, O(Lλ)). By Atiyah’s L index theorem (which we blackbox for this survey, see [10] for a brief survey), we have (after suitably normalizing the plancherel measure µ)

X X p p ∗,sm T (−1) · dim(H (n,Vi ) ⊗ Cλ) · µ(Vi) p Vi a d.s. with inf. char λ+ρ X Y hλ + ρ, αi = (−1)q (−1)p dim Hp(X, O(L )) = (−1)q λ hρ, αi p α∈∆+ dim(G/K) where q = 2 . The ﬁrst equality following from Atiyah and the second equality is im- mediate by applying the Borel–Weil–Bott theorem and the Weyl dimension formula for highest weight representations. Now applying Lemma 3.6 we get w(λ+ρ) X q Y hλ + ρ, αi X (w)e µ(Vi)θ(Vi) = (−1) T hρ, αi D Vi a d.s. with α∈∆+ w∈W inf. char λ+ρ This proves the assertion, since for singular λ + ρ, the right hand side is 0. What is left is the regular case. The theorem of Schmid will follow from the following p ∗,sm Lemma 3.10. Suppose that Vi is a discrete series, λ + ρ is regular such that H (n,Vi ) ⊗ T Cλ 6= 0. Then (1) p = p(λ + ρ) as deﬁned in the theorem. p ∗,sm T (2) dim H (n,Vi ) ⊗ Cλ = 1 p ∗,sm T 0 (3) H (n,Vi ) ⊗ Cλ0 6= 0 if and only if λ + ρ = w(λ + ρ) for some w ∈ Wc. The argument here is a bit involved, we will come back to it in a future draft. 14

3.3. Geometric realizations in coherent cohomology. Recall from the previous subsection that we have G-equivariant map

∗ ∗ H(2)(D, O(Lλ)) −→ H (D, O(Lλ)).

In this subsection we brieﬂy record some properties of the coherent cohomology without proof. We keep the notations from the previous subsection.

∗ Theorem 3.11 (Schmid). H (D, O(Lλ)) is a Frechet space, on which the action of G is admissible of ﬁnite length, and has inﬁnitesimal character λ + ρ.

Theorem 3.12 (Schmid). If λ + ρ is antidominant regular, then

∗ ∗ H(2)(D, O(Lλ)) ,→ H (D, O(Lλ)) is injective with dense image, and the two corresponding representations of G are inﬁnitesimally equivalent.

Another instance where the map from L2 to coherent cohomology induces an inﬁnitesimal equivalence on the underlying G-representations is the following case, which is of arithmetic interest. For the setup, note that the following conditions on ∆+ are equivalent • There exists a regular λ + ρ with λ ∈ Λ such that p(λ + ρ) = 0; • G/K is hermitian symmetric, and the natural map G/T = D(∆+) → G/K is holomorphic.

Theorem 3.13 (Schmid). Let ∆+ a positive root system satisfying the equivalent condition above 0 0 and λ + ρ a regular elements satisfying p(λ + ρ) = 0. Then H(2)(D, O(Lλ)) ,→ H (D, O(Lλ)) is injective with dense image, and are inﬁnitesimally equivalent as G-representations.

4. Limits of discrete series 4.0.1. Parametrization. As usual we have compact Cartan and maximal compact subgroups + + + T ⊂ K ⊂ G, and ﬁx ∆ = ∆c t ∆n , which determines a dominant chamber Dom. An important class of irreducible admissible representations beyond discrete series are limits of discrete series. They are certain representations π(λ + ρ, C) indexed by a pair of parameters (λ + ρ, C), where C is a Weyl chamber, and λ + ρ a singular Harish-Chandra parameter in

(X∗(T ) + ρ)sing ∩ C.

Two limits of discrete series π(λ0 + ρ, C0) and π(λ + ρ, C) are inﬁnitesimally equivalent if and 0 0 only if there is w ∈ Wc such that w(λ + ρ) = λ + ρ and w(C ) = C.

Deﬁnition 4.1. (1) A limit of discrete series π(λ + ρ, C) is holomorphic if C = Dom. 13 + (2) π(λ + ρ, C) is non-degenerate if it is ∆c-regular, in other words, if for any α ∈ ∆c , hλ + ρ, αi= 6 0. Otherwise it is called degenerate.

13Similarly, a discrete series π(λ + ρ) is holomorphic if λ + ρ ∈ Dom, note that in the discrete series = regular case, λ + ρ lies in a unique Weyl chamber. 15

4.0.2. Appearance in coherent cohomology of Shimura varieties. One of the most crucial feature for non-degenerate limits of discrete series (compared to the degenerate ones) is that they appear in coherent cohomology of certain Shimura varieties. This has been useful in arithmetic applications (in particular in the construction of Galois representations associated to those automorphic forms whose infinity components are non-degenerate limits, by establishing congruences with ones whose infinity components are discrete series). To be slightly more precise, let (G, X) be a Shimura datum, where G is a reductive group over × Q and X the conjugacy class of Deligne cocharacter h with h : ResC/R(Gm)(R) = C → G(R), satisfying Deligne’s axioms for Shimura varieties. Let K∞ = Stab(h) ⊂ G(R) and H∞ ⊂ K∞ the × 0,0 1,−1 maximal torus containing h(C ). The cocharacter h induces a filtration on gC = g ⊕ g ⊕ −1,1 0 0,0 1,−1 1 1,−1 ∗ g with Fil = g ⊕ g and Fil = g . The stabilizer of Fil is denoted by P ⊂ G(C) which is a parabolic subgroup. Let p be its Lie algebra.

Theorem 4.2. Suppose that G(R) is connected. Let π(λ + ρ, Dom) be a non-degenerate holomorphic limit of discrete series. Let Vλ+ρn−ρc be the highest weight representation of weight λ + ρn − ρc, then 0 ﬁn ∨ dim H (q,K∞; π(λ + ρ, Dom) ⊗ Vλ+ρn−ρc ) = 1 while all higher degree (q,K∞) cohomology vanishes. Remark 4.3. We end with several remarks

(1) Limits of discrete series do not appear in (g,K∞)-cohomology, so in some sense this is best we could hope for. (2) The requirement of π being holomorphic guarantees that the non-vanishing is at degree 0, non-degenerate non-holomorphic limits still show up in (q,K∞)-cohomology, but not in degree 0. (3) This implies that, if π = πf ⊗ π∞ is an automorphic representation of G(Af ) (with (g,K∞)-module π∞ a non-degenerate limit of discrete series as above), then there exists a G(Af ) equivariant embedding 0 ∨ πf ,→ H (Sh(G, X), V λ+ρn−ρc ). Note that degenerate limits of discrete series do not show up in coherent cohomology.

Appendix A. Various cohomology theories In this section we recall the deﬁnitions of relevant cohomology theories. Everything reviewed here is standard, maybe except for Franke’s theorem, for which a reference is included. A.1. Lie algebra cohomology. Let g be a Lie algebra over an arbitrary ﬁeld K and V a g representation (which is equivalent to a U(g)-module). Recall the following functor I = ( )g : g-Rep −→ Sets which sends V to V g = {v ∈ V : Xv = 0 for all X ∈ g}. Let K be the trivial 1-dimensional g representation (corresponding to the 0 map U(g) → K =∼ End(K)). Then we have ∼ g HomU(g)(K,V ) −→ I(V ) = V

In other words, the functor I is isomorphic to HomU(g)(K, −), which is a left exact functor from g-Rep = Mod/U(g) to Sets. Its right derived functors are the Lie algebra cohomology of V : p p ∼ p H (g,V ) = R I(V ) = ExtU(g)(K,V ). In order to compute this cohomology we need a projective resolution of K in the category of U(g)-modules. One (standard) way to do this is via the Chevalley–Eilenberg complexes. 16

Lemma A.1. The following complex provides a projective resolution of K in the category of U(g)-modules: · · · → ∧2 g ⊗ U(g) → g ⊗ U(g) → U(g) → K, where the diﬀerentials on ∧n g ⊗ U(g) −→d ∧n−1 g ⊗ U(g) are given by

n−1 X i (X0 ∧ · · · ∧ Xn−1) ⊗ U 7−→ (−1) X0 ∧ · · · ∧ Xbi ∧ · · · ∧ Xn−1 ⊗ XiU + i=0 X i+j (−1) [Xi,Xj] ∧ · · · Xbi ··· Xbj · · · ∧ Xn−1 ⊗ U 0≤i

n ∼ n n HomU(g)(∧ g ⊗ U(g),V ) −→ HomK (∧ g,V ) =: C (g,V ). For completeness, we write the diﬀerential on Cn(g,V ) as well: for each f ∈ Hom(∧ng,V ),

n−1 X i df(X0 ∧ · · · ∧ Xn−1) = (−1) Xi · f X0 ∧ · · · ∧ Xbi ∧ · · · ∧ Xn−1 + i=0 X i+j (−1) f [Xi,Xj] ∧ · · · Xbi ··· Xbj · · · ∧ Xn−1 . i

Remark A.2 (Motivation for Chevalley–Eilenberg complexes). 14 One may compare the complex defined above with the de Rham complex for a smooth manifold over R. Suppose that M is a real manifold, let X(M) = C∞(M,TM) be the space of smooth vector fields 15 on M. The set of differential k-forms Ωk(M) can be interpreted as alternating, C∞(M)-linear maps from X(M)k → C∞(M) 16. Now we may write the differential on the de Rham complex · · · → Ωn−1(M) −→d Ωn(M) → · · · via its identification

n ∼ n ∞ Ω (M) −→ HomC∞(M)(∧ X(M),C (M))

14Actually I am not entirely sure whether this was how Chevalley–Eilenberg came up with their specific resolution, I have not yet checked their paper (published 70 years ago). 15Vector fields are as usual defined as smooth sections of the tangent bundle. Via the identification between tangent vectors and directional derivatives, a vector field X corresponds to a smooth derivation X : C∞(M) → C∞(M). 16In other words, we claim that there is a natural bijection

k ∞ k ∗ ∼ k ∞ Ω (M) = C (M, ∧ T M) −−→ HomC∞(M)(∧ X(M),C (M)), given by ω 7→ ωe, which sends ∧X = X0 ∧ · · · ∧ Xk−1 to

ω×(∧X) k ∗ k ev ωe(∧X): M −−−−−→∧ T M ×M ∧ TM −→ R. Or, more explicitly ωe X0 ∧ · · · ∧ Xk−1 : p 7−→ ωp X0(p) ∧ · · · ∧ Xk−1(p) . Note that in the world of smooth manifolds, ∧kX(M) =∼ C∞(M, ∧kTM). 17

n−1 n as follows: for any ω ∈ Ω , X0 ∧ · · · ∧ Xn−1 ∈ ∧ X(M),

n−1 X i dω(X0 ∧ · · · ∧ Xn−1) = (−1) Xi · ω X0 ∧ · · · ∧ Xbi ∧ · · · ∧ Xn−1 + i=0 X i+j (−1) ω [Xi,Xj] ∧ · · · Xbi ··· Xbj · · · ∧ Xn−1 . i

Let ωG be a suitably normalized volume form on G, then we may deﬁne ∗ ∗ G ∗ ∗ ψ :Ω (M) → Ω (M) , by ω 7−→ πM,∗ πG ωG ∧ r ω .

Note that the push-forward along πM involves integration over the fiber of πM , which requires G to be compact. As a corollary we get Corollary A.4. For a compact connected real Lie group G, its de Rham cohomology can be computed by Lie algebra cohomology: p ∼ ∗ H (g, R) −→ HdR(G/R). A.2. L2 cohomology. Let M be a (not necessarily compact) smooth real manifold 17. To define L2 cohomology we consider differential forms that are square integrable. More precisely, let Ωi = Ωi(M) be the space of (C∞-sections of) i-forms, and L2(M)i be the L2 completion of Ωi. Define i i 2 Ωe(2)(M) := Ω (M) ∩ L (M) i i 2 i+1 Ω(2)(M) := {α ∈ Ωe(2)(M): dα ∈ L (M) }. Then one obtains the following complex 0 d 1 d 2 d 0 → Ω(2)(M) −→ Ω(2)(M) −→ Ω(2)(M) −→· · ·

17In fact one could work with manifold with singularities 18

2 i i ∗ whose cohomology is the L -cohomology of M: H(2)(M) = H Ω(2)(M) . We note that L2-cohomology is not topologically invariant. The simplest example is for the real line R with its standard metric, it is easy to see that 0 1 H(2)(R) = 0,H(2)(R) is infinite-dimensional. Next we introduce a variant of L2 cohomology. The strong closure d of the differential operator i i d is defined as follows: α ∈ Ω(2)(M) and dα = β if there exists a sequence αj ∈ Ω(2)(M) such 2 ∗ that αj −→ α and dαj −→ β in L . It turns out that using the complex Ω(2)(M) with the differential d, we get back the same cohomology for M. Now we take one step further, and consider the closure of the image of d, this defines for us the reduced L2 cohomology i H(2)(M) := ker di/im di−1. References [1] Harish-Chandra, Integrable and square-integrable representations of a semisimple Lie group. Proc. Nat. Acad. Sci. U.S.A. (1955). [2] Harish-Chandra, Representations of semisimple Lie groups IV, Amer. J. Math (1956) [3] Harish-Chandra, Invariant eigendistributions on a semisimple Lie algebra, I.H.E.S. pub. Math (1965) [4] Harish-Chandra, Discrete series for semisimple Lie groups I, Acta Math, (1965) [5] Harish-Chandra, Discrete series for semisimple Lie groups II, Acta Math, (1966) [6] Harish-Chandra, Two theorems on semi-simple Lie groups. Annals of Math. (1966) [7] Knapp, Representation theory of Semisimple Groups: an overview based on examples. Princeton Press. (1995) [8] Schmid, On a conjecture of Langlands. Annals of Math. (1971) [9] Schmid, L2-cohomology and the discrete series. Annals of Math. (1976) [10] Schmid, Discrete Series. Proc. Symp. Pure Math. (1997) [11] Wallach, Real reductive groups, vol. I. Academic Press. (1988)