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 2 Gb f understand πjK Questions K -theory (nice compact subgroup K ⊂ G). K -theory & repns Get an invariant of a repn π 2 Gb: mπ : Kb ! N; mπ(µ) = mult of µ in πjK . 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 2 Gb: Applications Pop up ad mπ : Kb ! N; mπ(µ) = mult of µ in πjK . 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 πjK (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 f j 2 g 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 = p2P aπ Sp (finite sum). 4. find algorithm to compute each Sp. (3) + (4) computes mπ, (2) gives meaning to answer. Today P = ftempered reps of real infl charg, 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 j = U(n) ( ) = ( ) ( 2 [( )n): π K IndU(1)n γ mµ γ γ γ U 1 Examples µ2U[(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 j = ( ) = ( ): Questions π K IndO(1)n γ mµ γ µ2O[(n) K -theory mπ(µ) = mult of µ in π is mµ(γ) = mult of γ in µjO(1)n . K -theory & repns 3. G split of type E8, K = Spin(16). Typical res to K is X j Spin(16) π Spin(16) = IndM (γ) = mµ(γ)γ; µ2Spin[(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 rq 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 rq 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 rq 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 rq q 2 qr (3; −2) (5; −2) q q q q q 1 rq q 2 qr q 3 (2; −3) (4; −3) (6; −3) q q q q q rq q 1 rq 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 rq 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 2 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 2 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 2 Kb λ 2 Tb. Precisely: genuine char of Teρ, M-regular. 4. Put P = MAN (cuspidal) parabolic, Questions K -theory δ 2 Mb discrete series with HC param λ. K -theory & repns ∗ G 5. ν 2 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) 2 Gb containing µ. 8. Every π 2 Gb of LKT µ is J(λ, ν)(µ), some ν 2 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.