This is what I do:

David Vogan

Introduction

Examples

Methods

This is what I do: Applications (restrict representations to compact subgroups) Pop up ad

Questions David Vogan K -theory K -theory & repns Department of Mathematics Massachusetts Institute of Technology

RTNCG Zoom 3/15/21 Outline This is what I do: David Vogan

Introduction Introduction Examples Examples Methods Applications How do you do that? Pop up ad

Applications Questions K -theory

And now a word from our sponsor K -theory & repns

What are the questions?

Equivariant K -theory

K -theory and representations Setting This is what I do: David Vogan

Introduction Compact groups K are relatively easy. . . Examples Methods Noncompact groups G are relatively hard. Applications Harish-Chandra et al. idea: Pop up ad

understand π ∈ Gb f understand π|K Questions K -theory

(nice compact subgroup K ⊂ G). K -theory & repns Get an invariant of a repn π ∈ Gb:

mπ : Kb → N, mπ(µ) = mult of µ in π|K .

1. What’s the support of mπ? (subset of Kb)

2. What’s the rate of growth of mπ? 3. What functions on Kb can be mπ? Where can you do this? This is what I do: David Vogan

Introduction

Examples

Methods

Get an invariant of a repn π ∈ Gb: Applications

Pop up ad mπ : Kb → N, mπ(µ) = mult of µ in π|K .

1. What’s the support of mπ? (subset of Kb) Questions 2. What’s the rate of growth of mπ? K -theory

3. What functions on Kb can be mπ? K -theory & repns I look at G = G(R) reductive, K = K (R) max cpt. Questions are just as interesting, and much less understood, for G p-adic, K compact open. What’s an answer look like? This is what I do: David Vogan

Introduction

Examples Seek to compute/understand mult of µ in π|K (π irr of G, µ irr of K ). Methods Applications

mπ is an N-valued function on an infinite countable set. Pop up ad One kind of answer: Questions { | ∈ } 1. find family Sp p P of functions Sp : bZ. K -theory 2. find nice interpretation of each Sp. K -theory & repns P p 3. find algorithm to write mπ = p∈P aπ Sp (finite sum).

4. find algorithm to compute each Sp.

(3) + (4) computes mπ, (2) gives meaning to answer.

Today P = {tempered reps of real infl char}, Sp = mp (branching rule for tempered p). Examples This is what I do: David Vogan 1. G = GL(n, C), K = U(n). Typical restriction to K is X Introduction | = U(n) ( ) = ( ) ( ∈ [( )n): π K IndU(1)n γ mµ γ γ γ U 1 Examples µ∈U[(n) Methods

mπ(µ) = mult of µ is mµ(γ) = dim of γ wt space. Applications Pop up ad 2. G = GL(n, R), K = O(n). Typical restriction to K is O(n) X | = ( ) = ( ): Questions π K IndO(1)n γ mµ γ µ∈O[(n) K -theory mπ(µ) = mult of µ in π is mµ(γ) = mult of γ in µ|O(1)n . K -theory & repns

3. G split of type E8, K = Spin(16). Typical res to K is X | Spin(16) π Spin(16) = IndM (γ) = mµ(γ)γ; µ∈Spin[(16) here M ⊂ Spin(16) subgp of order 512, central ext of (Z/2Z)8.

Moral: The hard work of computing mπ takes place inside the world of compact groups. Lowest K -types for Sp(4, R) This is what I do: David Vogan

Representations of maximal com- Introduction pact K = U(2) for G = Sp(4, R), Examples parametrized by pairs of integers q a ≥ b. Grouped in families as low- Methods est K -types of a particular series q q Applications x=2, hol. disc. series of representations, one for each (3, 3) q q q Pop up ad KGB element x with no complex descents. q q q q £ £ £ £ £ x=5, long Questions x (1, 1) q q q q q root Levi =10, min. K -theory princ. ser. q q q q q q (0, 0) (1, 0) K -theory & repns (−1,−1) (0,−1) q q ¢q ¡ ¢q(3,¡−1) ¢q ¡ ¢q ¡ ¢q ¡ @ x=0, non-hol. disc. series q q £q q ¡ q@ q q q ¢ ¡ £ £ £ ¡ @ q q q @q¢ q q q q q (−3,−3) ¢ ¡ (1,−3) ¢ ¡ @ @£ q £q q q q q¢ £q q ¡ q@ q ¢ ¡ @¢ ¡ @ £ £ £ ¡ q q ¢q q q ¡ q q @q¢ ¢q ¡ q q x=4, short £ @£ £ ¡ q q q ¢q q q ¡ q q q @q¢ ¢q ¡ q root Levi x=3, antihol. x=6, long x=1, non-hol. £ disc series £root Levi disc. series £ q q q q ¢q q q ¡ q q q q q¢ ¢q ¡ Branching law for ONE tempered rep This is what I do: David Vogan

Introduction In blue are K -types of the tempered rep- Examples resentation induced from the first holo- q 1 morphic discrete series on the long root (6, 5) Methods q 1 qr Levi. Changing first to kth again gives (5, 4) Applications an infinite triangle, bounded on the left q 1 qr q 2 Pop up ad by the line (k + 1, ∗) and above by the (4, 3) (6, 3) edge of the dominant chamber. q 1 qr q 2 qr (3, 2) (5, 2) Questions Multiplicities are shown in red next to q 1 qr q 2 qr q 3 each U(2)-representation. (1, 1) (2, 1) (4, 1) (6, 1) K -theory q qr q 1 qr q 2 qr (0, 0)(1, 0) (3, 0) (5, 0) K -theory & repns q q q 1 qr q 2 qr q 3 (−1,−1)(0,−1) (2, −1) (4, −1) (6, −1) q q q qr q 1 qr q 2 qr (3, −2) (5, −2) q q q q q 1 qr q 2 qr q 3 (2, −3) (4, −3) (6, −3) q q q q q qr q 1 qr q 2 qr (3, −4) (5, −4) q q q q q q q 1 qr q 2 qr q 3 (2, −5) (4, −5) (6, −5) q q q q q q q qr q 1 qr q 2 qr (3, −6) (5, −6) q q q q q q q q q qr q qr q How do you do that? This is what I do: David Vogan

Main ideas are due to Harish-Chandra, Langlands, Introduction

Schmid, and Knapp-Zuckerman. Examples Description below makes it sound like they’re due to me. Methods Reason: (given my age) I interpreted the topic as What I did. Applications Pop up ad Start with preorder on Kb, roughly length of highest weight. Every rep π ∈ Gb has one or more lowest K -types µ. Questions Study all G-reps π of lowest K -type µ. K -theory K -theory & repns Find: lowest K -type condition =⇒ Lie alg cohom(π) , 0. WHICH Lie alg cohom depends on how singular µ is. Gives the eight familes in picture of LKT’s for Sp(4, R). More generic µ better cohom fewer π with LKT µ. Most generic µ unique (“disc ser”) rep of G with LKT µ. Four red regions in Kb in Sp(4, R) picture. Least generic µ rk G-param fam (“princ ser”) reps with LKT µ. Black region of five K -types in Sp(4, R) picture. What do those methods tell you? This is what I do: David Vogan

Introduction Theorem (Langlands) Reductive G ⊃ K max compact. Examples Methods 1. µ ∈ Kb H(R) = TA ⊂ G Cartan subgroup. exp Applications = ∩ ' a 2. T H K compact, A 0 split vector group. Pop up ad 3. µ ∈ Kb λ ∈ Tb. Precisely: genuine char of Teρ, M-regular. 4. Put P = MAN (cuspidal) parabolic, Questions K -theory

δ ∈ Mb discrete series with HC param λ. K -theory & repns ∗ G 5. ν ∈ a , std rep I(λ, ν) = IndP (δ ⊗ ν ⊗ 1) 6. I(λ, ν) has µ as a LKT, multiplicity one. 7. I(λ, ν) has unique irr J(λ, ν)(µ) ∈ Gb containing µ. 8. Every π ∈ Gb of LKT µ is J(λ, ν)(µ), some ν ∈ a∗. 9. If ν = 0, µ is the unique LKT of J(λ, ν). Consequence: reps of LKT µ indexed by cplx vec space a∗. Composition series and characters This is what I do: David Vogan

Introduction

Prev slide exactly right; this slide needs care with lim disc ser. Examples Theory of lowest K types start with preorder on Kb. Methods Langlands classification gives bijection Kb ↔ (T , λ). Applications Pop up ad preorder on pairs (T , λ), inherited by params (TA, λ, ν). P λ0,ν0 0 0 Each std has finite comp series I(λ, ν) = λ0,ν0 m J(λ , ν ). Questions 0 0 λ,ν Nonnegative integer coeffs mλ ,ν , (λ0, ν0) ≥ (λ, ν). λ,ν 0 0 K -theory Equality in preorder only if (λ0, ν0) = (λ, ν), and then mλ ,ν = 1. λ,ν K -theory & repns P λ0,ν0 0 0 Each irr has finite char formula J(λ, ν) = λ0,ν0 Mλ,ν I(λ , ν ). λ0,ν0 0 0 Integer coeffs Mλ,ν , (λ , ν ) ≥ (λ, ν). 0 0 λ0,ν0 Equality in preorder only if (λ , ν ) = (λ, ν), and then Mλ,ν = 1. λ0,ν0 λ0,ν0 Matrices mλ,ν and Mλ,ν are integer, upper triang, 1s on diag. They are inverse to each other.

λ0,ν0 Mλ,ν is computed by Kazhdan-Lusztig theory. Restriction to K This is what I do: David Vogan

Introduction

Examples | Recall that plan to compute π K was to write multiplicity Methods function m as finite sum of nice functions M . π p Applications We’ve done that: Pop up ad

X λ0,ν0 0 0 π = Mπ I(λ , ν ) Questions λ0,ν0 K -theory X λ0,ν0 0 0 X λ0,ν0 0 π|K = Mπ I(λ , ν )|K = Mπ I(λ , 0)|K K -theory & repns λ0,ν0 λ0,ν0 X λ0 0 π|K = aπ I(λ , 0)|K . λ0 λ0 P λ0,ν0 0  Here aπ = ν0 Mπ , and {I(λ , 0)} = tempered, real infl char .

0 Remains to compute I(λ , 0)|K . A great story, not told today. What good is this? This is what I do: David Vogan

Introduction

Examples Original problem for inf-diml reps: which π ∈ Gb(R) are unitary? Methods Applications Knapp and Zuckerman: which π admit invt Hermitian form h, iπ. Pop up ad

If form exists, and K rep µ has multiplicity mπ(µ), then “form

restricted to µ gives signature (pπ(µ), qπ(µ)). Questions

K -theory Find in this way two new functions pπ, qπ from Kb to N. K -theory & repns 0 Like mπ, pπ, qπ = finite integer combs of I(λ , 0)|K . Compute signature ! compute finitely many integers. This sounds like a job for a computer. [Adams/van Leeuwen/Trapa/DV], Astérisque 417: it is. How do you compute these things really? This is what I do: David Vogan

Introduction A lot of this mathematics is twentieth century. Examples Around 2000, Jeffrey Adams set out to make Methods software implementing all these algorithms. Applications Pop up ad He succeeded, mostly because he managed to

interest Fokko du Cloux and Marc van Leeuwen. Questions Software is at http://atlas.math.umd.edu/. K -theory enter any real reductive G, any parameter p. K -theory & repns Then can type composition_series(p) character_formula(p) print_branch_irr(p,[height]) is_unitary(p) . . . and much more. Plan for today This is what I do: David Vogan

Introduction

Examples

Remaining slides (not presented): work with Jeff Adams Methods

to compute associated varieties of G representations. Applications

Pop up ad Work with real reductive Lie group G(R). Questions Describe (old) associated cycle AC(π) for irr rep [ K -theory π ∈ G(R): geometric shorthand for approximating K -theory & repns restriction to K (R) of π. Describe algorithm with Adams to compute AC(π). A real algorithm is one that’s been implemented on a computer. This one has been, by Adams in the atlas software. Assumptions This is what I do: David Vogan

Introduction G(C) = G = cplx conn reductive alg gp. Examples G(R) = group of real points for a real form. Methods Applications

Could allow fin cover of open subgp of G(R), so allow nonlinear. Pop up ad K (R) ⊂ G(R) max cpt subgp; K (R) = G(R)θ. θ = alg inv of G; K = Gθ possibly disconn reductive. Questions K -theory

Harish-Chandra idea: K -theory & repns ∞-diml reps of G(R) ! alg gp K y cplx Lie alg g (g, K )-module is vector space V with P 1. repn πK of algebraic group K : V = µ∈Kb mV (µ)µ 2. repn πg of cplx Lie algebra g −1 3. dπK = πg|k, πK (k)πg(X)πK (k ) = πg(Ad(k)X). In module notation, cond (3) reads k · (X · v) = (Ad(k)X) · (k · v). Geometrizing representations This is what I do: David Vogan G(R) real reductive, K (R) max cpt, g(R) Lie alg Introduction ∗ ∗ N = cone of nilpotent elements in g . Examples ∗ ∗ ∗ Methods NR = N ∩ ig(R) , finite # G(R) orbits. Applications N∗ = N∗ ∩ (g/k)∗, finite # K orbits. θ Pop up ad Goal 1: Attach orbits to representations in theory. Goal 2: Compute them in practice. Questions K -theory “In theory there is no difference between theory and practice. In K -theory & repns practice there is.” Jan L. A. van de Snepscheut (or not). K (π, Hπ) irr rep of G(R) H irr (g, K )-module −→ π−→ Howe wavefront assoc var of gr ∗ ∗ WF(π) = G(R) orbs on NR AC(π) = K orbits on Nθ

Columns related by HC, Kostant-Rallis, Sekiguchi, Schmid-Vilonen. So Goal 1 is completed. Turn to Goal 2... Associated varieties This is what I do: David Vogan F (g, K ) = finite length (g, K )-modules. . . noncommutative world we care about. Introduction Examples C(g, K ) = f.g. (S(g/k), K )-modules, support ⊂ N∗... θ Methods

commutative world where geometry can help. Applications

gr Pop up ad F (g, K ) C(g, K )

gr not quite a functor (choice of good filts), but Questions Prop. gr induces surjection of Grothendieck groups K -theory gr K -theory & repns K F (g, K ) −→ K C(g, K ); image records restriction to K of HC module. So restrictions to K of HC modules sit in equivariant coherent sheaves on nilp cone in (g/k)∗

K ∗ K C(g, K ) =def K (Nθ ), equivariant K -theory of the K -nilpotent cone. K ∗ Goal 2: compute K (Nθ ) and the map Prop. Equivariant K -theory This is what I do: David Vogan

Setting: (complex) algebraic group K acts on Introduction

(complex) algebraic variety X. Examples Originally K -theory was about vector bundles, but for Methods us coherent sheaves are more useful. Applications Pop up ad CohK (X) = abelian categ of coh sheaves on X with K action. K K K (X) =def Grothendieck group of Coh (X). Questions Example: CohK (pt) = Rep(K ) (fin-diml reps of K ). K -theory K -theory & repns K K (pt) = R(K ) = rep ring of K ; free Z-module, basis Kb. Example: X = K /H; CohK (K /H) ' Rep(H)

E ∈ Rep(H) E =def K ×H E eqvt vector bdle on K /H K K (K /H) = R(H). Example: X = V vector space. K E ∈ Rep(K ) proj module OV (E) =def OV ⊗ E ∈ Coh (X) K  proj resolutions =⇒ K (V ) ' R(K ), basis OV (τ) . Doing nothing carefully This is what I do: David Vogan Suppose K y X with finitely many orbits: Introduction yi X = Y1 ∪ · · · ∪ Yr , Yi = K · yi ' K /K . Examples

Orbits partially ordered by Yi ≥ Yj if Yj ⊂ Yi . Methods Applications K (τ, E) ∈ Kdyi E(τ) ∈ Coh (Y ). i Pop up ad Choose (always possible) K -eqvt coherent extension

K K Questions Ee(τ) ∈ Coh (Yi ) [Ee] ∈ K (Yi ). K -theory

K K -theory & repns Class [Ee] on Y i unique modulo K (∂Yi ). K Set of all [Ee(τ)] (as Yi and τ vary) is basis of K (X). Suppose M ∈ CohK (X); write class of M in this basis Xr X [M] = nτ(M)[Ee(τ)].

i=1 τ∈Kdyi

Maxl orbits in Supp(M)= maxl Yi with some nτ(M) , 0.

Coeffs nτ(M) on maxl Yi ind of choices of exts Ee(τ). Our story so far This is what I do: David Vogan

We have found Introduction 1. homomorphism Examples Methods gr K ∗ virt G(R) reps K F (g, K ) −→ K (Nθ ) eqvt K -theory Applications Pop up ad n o K ∗ 2. geometric basis [Eg(τ)] for K (Nθ ), indexed by irr reps of isotropy gps Questions K -theory 3. expression of [gr(π)] in geom basis AC(π). K -theory & repns Problem is expressing ourselves... Teaser for the next section: Kazhdan and Lusztig taught us how to express π using std reps I(γ): X [π] = mγ(π)[I(γ)], mγ(π) ∈ Z. γ  K ∗ [gr I(γ)] is another basis of K (Nθ ). Last goal is compute change of basis matrix. The last goal This is what I do: David Vogan ∗ Studying cone Nθ = nilp lin functionals on g/k. Introduction n o Found (for free) basis [Eg(τ)] for K K (N∗), indexed by Examples θ Methods i i orbit K /K and irr rep τ of K . Applications  Found (by rep theory) second basis [gr I(γ)] , Pop up ad indexed by (parameters for) std reps of G(R). To compute associated cycles, enough to write Questions K -theory

X X K -theory & repns [gr I(γ)] = Nτ(γ)[Ee(τ)]. orbits τ irr for isotropy Equivalent to compute inverse matrix X [Ee(τ)] = nγ(τ)[gr I(γ)]. γ Need to relate geom of nilp cone to geom std reps: parabolic subgroups. Use Springer resolution. Introducing Springer This is what I do: David Vogan ∗ g = k ⊕ s Cartan decomp, Nθ 'Nθ =def N ∩ s nilp cone in s. Introduction Kostant-Rallis, Jacobson-Morozov: nilp X ∈ s Y ∈ s, H ∈ k Examples [H, X] = 2X, [H, Y ] = −2Y , [X, Y ] = H, Methods g k ⊕ s [k] = [k] [k] (ad(H) eigenspace). Applications g[≥ 0] = q = l + u θ-stable parabolic. def Pop up ad Theorem (Kostant-Rallis) Write O = K · X ⊂ Nθ. Questions 1. µ: OQ =def K ×Q∩K s[≥ 2] → O, (k, Z) 7→ Ad(k)Z is proper birational map onto O. K -theory K -theory & repns 2. K X = (Q ∩ K )X = (L ∩ K )X (U ∩ K )X is a Levi X decomp; so KdX = [(L ∩ K ) ]b. So have resolution of singularities of O: K ×Q∩K s[≥ 2] vec bdle .& µ K /Q ∩ K O Use it (i.e., copy McGovern, Achar) to calculate equivariant K -theory. . . Using Springer to calculate K -theory This is what I do: David Vogan

Introduction

Examples X ∈ Nθ represents O = K · X. Methods

µ: OQ =def K ×Q∩K s[≥ 2] → O Springer resolution. Applications X Pop up ad Theorem Recall KdX = [(L ∩ K ) ]b . K 1. K (OQ) has basis of eqvt vec bdles: Questions (σ, F) ∈ Rep(L ∩ K ) F (σ). K -theory K -theory & repns 2. Get extension of E(σ|(L∩K )X ) on O X i i K [F (σ)] =def (−1) [R µ∗(F (σ))] ∈ K (O). i P 3. Compute (very easily) [F (σ)] = γ nγ(σ)[gr I(γ)]. X 4. Each irr τ ∈ [(L ∩ K ) ]b extends to (virtual) rep σ(τ) of L ∩ K ; can choose F (σ(τ)) as extension of E(τ). Now we’re done This is what I do: David Vogan X Recall X ∈ Nθ O = K · X; τ ∈ [(L ∩ K ) ]b . Introduction Now we know formulas Examples X Methods [Ee(τ)] = [F (σ(τ))] = nγ(τ)[gr I(γ)]. Applications γ Pop up ad Here’s why this does what we want: Questions 1. inverting matrix nγ(τ) matrix Nτ(γ) writing [Ee(τ)] in K -theory terms of [gr I(γ)]. K -theory & repns

2. multiplying Nτ(γ) by Kazhdan-Lusztig matrix mγ(π) matrix nτ(π) writing [gr π] in terms of [Ee(τ)].

3. Nonzero entries nτ(π) AC(π). Side benefit: algorithm (for G(R) cplx) also computes bijection (conj by Lusztig, estab by Bezrukavnikov)

(dom wts) ↔ (pairs (τ, O))