Unitary representations of

reductive Lie groups

David Vogan May 25, 1999

Outline

Gelfand’s abstract harmonic analysis Quantum mechanics and Hilbert spaces Classical mechanics and symplectic geometry Problem: Unitary repns of simple Lie groups Classical repn theory: Coadjoint orbits Problem: Quantization for coadjoint orbits Three kinds of coadjoint orbit and two kinds of quantization Towards a third kind of quantization

Abstract Harmonic Analysis

Setting: G (topological) group acting on X (topological) space; class of “interesting questions” about X

Step 1. Attach to X a (topological) vector space V, so questions about X become questions about V.

Step 2. Find (finest possible) G-equivariant decomposition V = ∑_{i} V_{i}. Hope each V_{i} is an irreducible representation of G.

Step 3. Understand each irreducible representation of G. (Topic for today!)

Step 4. Assemble information to answer questions about X.

First example

G = S_{n}, symmetric group on n letters; X = p-element subsets of {1, …, n}.

Question: what is the cardinality of X? Step 1. V = functions on X; card X = dim V. Step 2. First decompose

V = ∑_{τ partition of n} m_{τ}V_{τ}.

Say p ≤ n − p. Then m_{τ} = 0 unless τ = (r, s) with

s ≤ p (when m_{τ} = 1). Have

dim V = ∑_{τ} m_{τ} dim V_{τ}

Step 3. Calculate

dim V_{τ} = # std Young tableaux of shape τ

dim V_{(r,s)} = \binom{n}{s} (\frac{r − s + 1}{r + 1} )

Step 4. Conclude (for p ≤ n − p)

card(X) = ∑_{s=0}^{p} \binom{n}{s} (\frac{n − 2s + 1}{n − s + 1} ) = \binom{n}{p}

Second example

G = real reductive Lie ⊃ Γ discrete cocompact X = G/Γ, Z = loc. symm. space K\G/Γ

Problem: understand de Rham cohomology H^{p}(Z). Step 1. V = C^{∞}(X); H^{p}(Z) ≃ H^{p}(𝔤, K; V) (relative

Lie algebra cohomology). Step 2. First decompose

V = ∑_{π∈_{G}}m_{π}(Γ)V_{π}.

Sum is over irr. unitary reps. π of G. Have

H^{p}(Z) = ∑_{π} m_{π}(Γ) dim H^{p}(𝔤, K; V_{π})

Step 3. Classify irr. unitary reps. π with H^{p}(𝔤, K; V_{π}) not zero. Main ex. is V_{π} = ℂ, H^{p}(𝔤, K; ℂ) ≃ H^{p}(U), U = cpt. dual of G/K. Other π ⟷ smaller cpt. symm. space U_{π},

H^{p}(𝔤, K; V_{π}) ≃ H^{p+d_{π}} (U_{π}), d_{π} = \frac{1}{2}(dim U − dim U_{π}).

Step 4. Conclude H^{p}(K\G/Γ) ≃ H^{p}(U), (p ≤ min_{π≠1} d_{π}).

Quantum mechanics

Physical system ⟷ complex Hilbert space H States ⟷ lines in H Observables ⟷ operators {A_{j}} on H Expectation of observable on state v ⟷ ⟨Av, v⟩ Energy ⟷ skew-adjoint operator A_{0} Time evolution ⟷ unitary gp exp (tA_{0}) Evolution of A ⟷ exp (−tA_{0})A exp (tA_{0}) Observable A conserved ⟷ [A_{0}, A] = 0

Classical mechanics

Symplectic manifold is manifold M with Lie algebra structure {, } on C^{∞}(M) satisfying

{a, bc} = {a, b}c + b{a, c}

and a nondegeneracy condition. Function f defines Hamiltonian vector field ξ_{f} = {f, ·}. Physical system ⟷ symplectic manifold M States (pos + velocities) ⟷ points in M Observables ⟷ smooth fns {a_{j}} ⊂ C^{∞}(M) Value of observable a on state m ⟷ a(m) Energy ⟷ real-valued function a_{0} Time evolution ⟷ flow of vector field ξ_{a_{0}} Evolution of a ⟷ da/dt = {a_{0}, a} Observable a conserved ⟷ {a_{0}, a} = 0

Unitary reps of Lie group G

A unitary representation of G is a Hilbert space H_{π} endowed with a continuous action

G × H_{π} → H_{π}, (g, v) ↦ π(g)v resp. inner product: ⟨v, w⟩ = ⟨π(g)v, π(g)w⟩. An invariant subspace is closed subspace V preserved by action; then V^{⊥} is invariant, and H_{π} = V ⊕ V^{⊥}. Say π is irreducible if it has exactly two invt subspaces. Unitary dual problem: Determine the set Ĝ_{u} of irr unitary reps of G. Want to emphasize the analogy unitary reps ⟷ quantum mech systems Put 𝔤_{0} = Lie(G), Lie alg of G. From X ∈ 𝔤_{0} get skew-adjt op dπ(X): π(tX) = exp (tdπ(X)). skew-adjoint ops {dπ(X)} ⟷ observables

“Classical” representations

after Kirillov and Kostant

A Hamiltonian G-space is a symplectic manifold M with smooth action

G × M → M, (g, m) ↦ g · m resp. symplectic structure. Each X ∈ 𝔤_{0} defines vector fld ξ(X) on M. Require also Geqvt Lie algebra homomorphism

𝔤_{0} → C^{∞}(M), X ↦ f(X)

satisfying ξ(X) = ξ_{f(X)} = {f(X), ·}. Classical analogue of “irreducible” is that M is homogeneous. Analogy here is Ham. G-spaces ⟷ classical mech systems functions f(X) ⟷ classical observables

Classification of “classical” reps

Define 𝔤_{0}^{∗} = dual space of Lie algebra. Elements of 𝔤_{0} are (linear) functions on 𝔤_{0}^{∗}; Lie bracket on these elements extends to Poisson bracket {, } on C^{∞}(𝔤_{0}^{∗}).

Fix M ⊂ 𝔤_{0}^{∗} orbit of G (a coadjoint orbit). Poisson bracket on 𝔤_{0}^{∗} restricts to (G-invt) symplectic structure on M. Elements X ∈ 𝔤_{0} restrict to functions f(X) on M; makes M = G/H a homogeneous Hamiltonian G-space. Structure is inherited by any equivariant covering space M̃ = G/H_{1}, with H ⊃ H_{1} ⊃ H_{0}.

Theorem (Kirillov-Kostant) Each homogeneous Hamiltonian G-space is a G-equivariant cover of a coadjoint orbit.

Philosophy of coadjoint orbits

Diagram of analogies and wishful thinking:

irr unitary reps  ⟷ quant mech
↑  ↑
coadjt orbit covers  ⟷ classical mech

Right vert arrow is “quantization;” should exist under some integrality constraint on classical system (since the world exists, and is quantummechanical).

Left vert arrow is the wishful thinking part: should exist by analogy with right.

Quantization problem: construct unitary reps from (suitably integral) coadjt orbits.

This won’t produce all irreducible unitary reps, but it does better than any other known approach. (Second prize to Arthur’s conjectures, based on Langlands’ philosophy and automorphic repn theory.)

Three kinds of coadjoint orbit…

Nice reductive Lie group G has nice map

G → GL(n, ℝ)

with discrete kernel. Leads to nice inclusion

𝔤_{0}^{∗} ⊂ n × n real matrices .

Element λ ∈ 𝔤_{0}^{∗} is called… • hyperbolic if corr matrix is diagonalizable • elliptic if diag/ℂ, purely imag eigenvalues • nilpotent if corr matrix is nilpotent

General coadjt orbits built in a simple way from these three types.

… and two kinds of quantization

Write M = G · λ ≃ G/G^{λ}, coadjt orbit.

Theorem If λ hyperbolic, there is a G-eqvt fibration M → Z with Lagrangian fibers and Z compact.

Can attach unitary rep to M, on space of L^{2} sections of a line bundle on Z.

Theorem If λ elliptic, there is G-eqvt cplx structure on M making M (indefinite) Kähler.

Under integrality constraint, can attach unitary rep to M, on Dolbeault cohomology of M with coeffs in holomorphic line bundle.

Theorem If λ nilpotent, then M is a cone (closed under positive dilations in 𝔤_{0}^{∗}).

Don’t know how to attach unitary rep to a nilpotent coadjoint orbit.

Quantizing nilpotent orbits

Define N_{ℝ}^{∗} = cone of nilpotent elements in 𝔤_{0}^{∗}. Finitely union of orbits of G (still a reductive group). Want to attach unitary reps to these orbits. Two guiding principles…

First, G has maximal compact subgroup K; understand unitary representations of K very well. Arise from “classical” picture using invt complex structure. (Example: Borel-Weil theorem.) Such reps extend naturally to complexification K_{ℂ} of K.

Idea: replace N_{ℝ}^{∗} with a complex algebraic variety N_{θ}^{∗} carrying algebraic K_{ℂ} action, so that actions of K on N_{ℝ}^{∗} and N_{θ}^{∗} are (more or less) equiv.

Construction: 𝔤^{∗} = cplx dual of 𝔤_{0},

N_{θ}^{∗} = {λ ∈ 𝔤^{∗} | λ |_{𝔨}= 0 and λ nilpotent }.

Kostant-Rallis: this is a complex algebraic variety on which K_{ℂ} acts with finitely many orbits.

Sekiguchi: orbits in 1–1 corr with G orbits on N_{ℝ}^{∗}

Vergne: corr orbits are K-eqvtly diffeomorphic

Conclusion: reps attached to nilpotent orbits should be realized (as reps of K) as (global sections of) K_{ℂ}-eqvt sheaves of modules on N_{θ}^{∗}.

The classical limit

Second guiding principle: so far considered only constructing quantum system from classical one. Physics also considers “classical limit:” let Planck’s constant go to zero. Amounts to making operators more commutative. Example is symbol calculus for diff ops on mfld X, replacing D(X) by C^{∞}(T^{∗}(X)).

Idea: use Poincaré-Birkhoff-Witt isom gr U(𝔤) ≃ S(𝔤). Try to replace reps (modules for U(𝔤)) by modules for poly ring S(𝔤).

Construction: V irr HC module for reductive G. Choose K-invt good filtration

V_{0} ⊂ V_{1} ⊂ ⋯ _{n}^{⋃} V_{n} = V U_{p}(𝔤) · V_{q} ⊂ V_{p+q} .

Then gr V is fin gen K_{ℂ}-eqvt S(𝔤/𝔨)-module supported on N_{θ}^{∗}.

Conclusion: reps V attached to nilpotent orbits should have gr V an uncomplicated module with specified supp in N_{θ}^{∗}.

Unitarity

Given nilpotent orbit, seek unitary rep of G so corr HC module V makes gr V an uncomplicated module with specified support. Kazhdan-Lusztig conjectures show how to construct many HC modules V′ with specified support. Two (related) problems: gr V′ may be complicated, and V′ may not be unitary.

Manifestation of first problem: multiplicity of gr V′ (along generic part of support) may be bigger than 1.

Manifestation of second problem: V′ may carry Hermitian form with indefinite signature.

Idea: show that second problem forces first to occur. Rule out first problem by algebra (for special V), and conclude that such V come from unitary representations.

Signatures

Theorem Suppose V irr HC module for reductive G admitting invt Hermitian form. Then there are K_{ℂ}-eqvt fg graded S(𝔤/𝔨)-modules M^{±} supp on N_{θ}^{∗}, with following properties:

1. gr V = M^{+} + M^{−} in Grothendieck group.

2. generic mult (gr V) = gen mult (M^{+} )+gen mult (M^{−}).

3. M^{−} = 0 iff form on V is positive definite.

4. There are virtual HC modules V^{±} satisfying

(a) gr V^{±} = M^{±} in Grothendieck group.

(b) |infl char (V^{±})| ≤ |infl char (V)|.

Conclusion: if generic multiplicity of gr V is 1, then support of gr V^{−} must be smaller than support of gr V.

Theorem Suppose V virtual HC module, and supp gr V is small. Then infl char (V) is large.

Problem: can’t yet make “large” and “small” precise.

Conclusion: if generic multiplicity of gr V is 1, and infl char (V) is small, then form on V is positive definite; so V comes from unitary rep of G.