Kirillov’s character formula, the holomorphic Peter-Weyl theorem, and the Blattner-Kostant-Sternberg pairing

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Kirillov’s character formula, the

holomorphic Peter-Weyl theorem, and the Blattner-Kostant-Sternberg pairing

J. Huebschmann

USTL, UFR de Math´ematiques CNRS-UMR 8524

59655 Villeneuve d’Ascq C´edex, France [email protected]

Krakow, July 3, 2008

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Abstract

By means of the orbit method we show that, for a compact Lie group, the Blattner-Kostant-Sternberg pairing map, with the constants being appropriately fixed, is unitary. Along the way we establish a holomorphic Peter-Weyl theorem for the complexifica- tion of a compact Lie group. Among our crucial tools is Kirillov’s character formula. The basic observation is that the Weyl vector is lurking behind the Kirillov character formula as well as behind the requisite half-form correction on which the Blatter- Kostant-Sternberg-pairing for the compact Lie group relies and thus produces the appropriate shift which, in turn, controls the unitarity of the BKS-pairing map. Our methods are independent of heat kernel harmonic analysis, which has been used by B. C.

Hall to obtain a number of these results. The heat kernel analysis comes out of our approach as well.

These results yield the first steps within a joint research program with G. Rudolph and M. Schmidt aimed at developing lattice gauge models for singular quantum mechanics. Within this program, we have already discovered a tunneling effect between singular strata.

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Contents

1 Motivation 4

2 Polar decomposition and K¨ahler struc-

ture 7

3 Cartan-Weyl decomposition 8 4 Holomorphic Peter-Weyl theorem 10 5 Abstract identification of the two Hilbert

spaces 12

6 BKS pairing 13

7 Kirillov’s character formula 14

8 Energy quantization 15

9 Relationship with heat equation 17

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1 Motivation

Q = Rⁿ, T^∗Q ∼= Cⁿ

z = q + ip holomorphic variables Segal-Bargmann :

ε Liouville measure, κ(q + ip) = p² HL²(Cⁿ, e^−κ/^~ε) ↔ L²(Rⁿ, dq)

— Lattice gauge theory: K compact Lie Lie algebra k with invariant inner product

T^∗K ∼= TK −→ K × k −→ K^C

— complex structure on K^C

— cotg. bdle symplectic structure on T^∗K combine to K-bi-invariant K¨ahler structure

Q = K^`, T^∗Q = T^∗K^` ∼= (K^C)^` K-symmetry by conjugation

reduced space

T^∗K^`

K ∼= (K^C)^`

K^C

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Quantization:

costratified Hilbert space, defined in terms of Hilbert space HL²(K^C, e^−κ/^~ηε) holomorphic functions on K^C square-integrable relative to e^−κ/^~ηε

κ and η to be defined shortly

Hilbert space HL²(K^C, e^−κ/^~ηε) “sees the strata”

We do not know how ordinary Hilbert space L²(K, dx) could possibly see the strata.

Need L²(K, dx) to understand observables B. Hall: coherent state transform

HL²(K^C, e^−κ/^~ηε) ↔ L²(K, dx) comes down to BKS pairing map and in

particular yields unitarity of BKS pairing map relies on heat kernel harmonic analysis

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Today: Holomorphic Peter-Weyl theorem for the Hilbert space

HL²(K^C, e^−κ/^~ηε) Unitarity of BKS pairing map

HL²(K^C, e^−κ/^~ηε) ↔ L²(K, dx) a direct consequence

new approach to coherent state transform independent of heat kernels

heat kernel analysis comes out for free

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2 Polar decomposition and K¨ahler structure

General compact Lie group K

Lie algebra k with invariant inner product T^∗K ∼= TK −→ K × k −→ K^C global K¨ahler potential κ

κ(xe^iY ) = |Y |², x ∈ K, Y ∈ k:

symplectic structure on T^∗K ∼= K^C: i∂∂κ ε symplectic volume form on T^∗K ∼= K^C η the real K-bi-invariant function on K^C

η(x e^iY ) =

s

sin(ad(Y )) ad(Y )

, x ∈ K, Y ∈ k half-form K¨ahler quantization on K^C:

holomorphic Hilbert space HL²(K^C, e^−κ/^~ηε) functions square-integrable relative to e^−κ/^~ηε scalar product

hψ₁, ψ₂i = 1 vol(K)

Z

K^C

ψ₁ψ₂e^−κ/^~ηε

left and right translation: HL²(K^C, e^−κ/^~ηε) unitary (K × K)-representation

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3 Cartan-Weyl decomposition

C[K^C] representative functions on K^C algebra via convolution product

Kd^C isomorphism classes of irred reps of K^C maximal torus in K

dominant Weyl chamber C⁺

highest weight λ: T_λ: K^C → End(V_λ) representation in the class of λ

ψ ∈ V_λ^∗, w ∈ V_λ

representative function Φ_ψ,w given by Φ_ψ,w(q) = ψ(qw), q ∈ K^C association

V_λ^∗ ⊗ V_λ 3 ψ ⊗ w 7−→ Φ_ψ,w morphism of (K^C × K^C)-reps

ι_λ: V_λ^∗ ⊗ V_λ −→ C[K^C]

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irreducible representation T_λ: K → End(W_λ) Fourier coefficient fb_λ ∈ End(W_λ) of

L²-function f on K relative to λ fb_λ = 1

vol(K) Z

K

f(x)T_λ(x⁻¹)dx irreducible rational representation

T_λ: K^C → End(V_λ) of K^C

Fourier coefficient Φb_λ ∈ End(V_λ) of holomorphic function Φ on K^C:

Fourier coefficient of the restriction of Φ to K F_λ: C[K^C] −→ End(V_λ), Φ 7→ Φb_λ

Theorem 3.1. [Cartan-Weyl]

(i) As (K^C × K^C)-representations C[K^C] = ⊕_λV_λ^∗ ⊗ V_λ

(ii) V_λ^∗ ⊗ V_λ isotypical summand of C[K^C] (iii) relative to convolution product on C[K^C],

(F_λ) : C[K^C] −→ ⊕_λEnd(V_λ) algebra isomorphism

isomorphism of (K^C×K^C)-representations

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4 Holomorphic Peter-Weyl theorem

R⁺ positive roots

Weyl vector: ρ = ¹₂ P

α∈R⁺ α

C_t,λ = (tπ)^dim(K^)/2e^t|λ+ρ|²

on End(V_λ) inner product h · , · i_λ given by hA, Bi_λ = tr(A^∗B), A, B ∈ End(V_λ)

adjoint A^∗ of A being computed as usual with respect to a K-invariant inner product on V_λ d_λ dimension of V_λ

endow ⊕

λ∈Kd^CEnd(V_λ) with the inner product which, on the summand End(V_λ), is given by

d_λ

C_t,λh · , · i_λ

up to a constant familiar Hilbert-Schmidt norm

⊕b

λ∈Kd^CEnd(V_λ)

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Theorem 4.1.[Holomorphic Peter-Weyl thm]

(i) HL²(K^C, e^−κ/tηε) contains vs C[K^C] of represent. fts on K^C as dense subspace

as unitary (K × K)-representation, HL²(K^C, e^−κ/tηε) decomposes as

HL²(K^C, e^−κ/tηε) = ⊕b

λ∈Kd^CV_λ^∗ ⊗ V_λ into (K × K)-isotypical summands

(ii) convolution induces a convolution product ∗ on HL²(K^C, e^−κ/tηε)

(iii) relative to convolution, as λ ranges over the irreducible rational representations of K^C, the assignment to a holomorphic

function Φ on K^C of its Fourier coefficients Φb_λ ∈ End(V_λ) yields an isomorphism

HL²(K^C, e^−κ/tηε) −→ ⊕b

λ∈Kd^CEnd(V_λ) of Hilbert algebras

Consequence of holomorphic Peter-Weyl: irreducible characters χ^C_λ of K^C a basis of

H = HL²(K^C, e^−κ/^~ηε)^K

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5 Abstract identification of the two Hilbert spaces

R(K) algebra of representative fn’s on K

W_λ unitary K-representation arising from V_λ by restriction to K

ι_λ: W_λ^∗ ⊗ W_λ −→ R(K) = C[K^C] Theorem 5.1. The association

V_λ^∗ ⊗ V_λ −→ W_λ^∗ ⊗ W_λ

ϕ^C 7−→ C_t,λ^1/2ϕ = (tπ)^dim(K^)/4e^t|λ+ρ|²^/2ϕ as λ ranges over the highest weights induces a unitary isomorphism

H_t: HL²(K^C, e^−κ/tηε) −→ L²(K, dx) of unitary (K × K)-representations.

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6 BKS pairing

Theorem 6.1. Above iso coincides with BKS-pairing map

H_t: HL²(K^C, e^−κ/tηε) −→ L²(K, dx) Φ 7−→ F_Φ

where, for x ∈ K, (F_Φ)(x) =

Z

k

Φ(x exp(iY ))e⁻

|Y |2

2t η(Y /2)dY.

K abelian: measure on K^C Haar measure on K times Gaussian in imaginary directions

decomposition of the Hilbert spaces into irreducible (1-dimensional) constituents:

inspection of characters establishes unitarity of BKS-pairing map

our argument for general K the same, with shift by Weyl vector incorporated, points in dual of abelian Lie algebra that correspond to characters replaced with entire integral

coadjoint orbits

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7 Kirillov’s character formula

Real analytic function j : K^C ∼= K ×k −→ R j(x, Y ) = j(Y ) = det

sinh(ad(Y /2)) ad(Y /2)

¹₂

obvious identity

j(iY ) = η(Y /2) highest weight λ

ϑ variable on Ω_λ+ρ = Ad^∗(K)(λ + ρ) ⊆ k^∗ Kirillov’s character formula:

vol(Ω_ρ)j(X)χ_λ(expX) = Z

Ω_λ+ρ

e^iϑ(X⁾dσ In our approach, this formula is used to

evaluate integrals, e. g.:

Lemma 7.1. ϕ^C in V_λ^∗ ⊗ V_λ Z

ϕ^Cϕ^Ce^−κ/tηε = C_t,λ Z

ϕϕdx

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8 Energy quantization

K¨ahler: only constants quantizable

Schr¨odinger: functions at most quadratic in generalized momenta quantizable

(classical) Hamiltonian of model quantizable associated quantum Hamiltonian

H = −~²

2 ∆_K + potential

The operator ∆_K arises from the non-positive Laplace-Beltrami operator associated with bi-invariant Riemannian metric on K. Since metric bi-invariant, so is ∆_K, whence

∆_K restricts to self-adjoint operator on L²(K,dx)^K

By means of

L²(K, dx) −→ HL²(K^C, e^−κ/tηε) transfer Hamiltonian, in particular, the operator ∆_K, to self-adjoint operators on

HL²(K^C, e^−κ/tηε)

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Schur’s lemma:

— each isotypical (K × K)-summand

L²(K, dx)_λ of L²(K, dx) in the Peter-Weyl decomposition an eigenspace

— representative functions eigenfunctions for

∆_K

— eigenvalue −ε_λ of ∆_K corresponding to the highest weight λ

ε_λ = (|λ + ρ|² − |ρ|²),

— in holomorphic quantization on T^∗K ∼= K^C, energy operator arises as the unique ex- tension of the operator −¹₂∆_K on

to an unbounded self-adjoint operator

— spectral decomposition thereof refines to holomorphic Peter-Weyl decomposition of

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9 Relationship with heat equation

p_t(x) = u(t, x) = X

λ

d_λe⁻^ελ² ^tχ_λ(x) fundamental solution of heat equation

du

dt = 1

2∆_Ku

on K, subject to initial condition p₀ Dirac distribution supported at identity of K

Under inverse transformation

L²(K, dx) −→ HL²(K^C, e^−κ/tηε)

up to a constant, a smooth function f on K goes to unique holomorphic function on K^C whose restriction to K given by

e^−t|ρ|²^/2e^t∆^K^/2f

for y ∈ K value (e^t∆^K^/2f)(y) given by (e^t∆^K^/2f)(y) =

Z

K

p_t(yx⁻¹)f(x)dx = (p_t∗f)(y), cf. [Nelson]. This establishes the relation

between the BKS-pairing and the heat equation observed by B. Hall.

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References

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[10] J. Huebschmann, Lie-Rinehart algebras, descent, and quantization, in: Galois theory, Hopf algebras, and semiabelian categories, Fields Institute Communications 43 (2004),295–316, Amer. Math. Society, Providence, R. I., math.SG/0303016

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[12] J. Huebschmann, G. Rudolph, and M. Schmidt, A gauge model for quantum mechanics on a stratified space, hep-th/0702017