Notes on microlocal aspects of representation theory Paul Nelson March 4, 2018 Contents 1 Overview 2 2 Lecture summaries and homework 2 2.1 2/21 and 2/22: Kirillov formula for SU(2) .............2 3 Prerequisites 5 4 Some background 5 4.1 Peter–Weyl theorem . .5 4.2 Symplectic manifolds . .5 4.2.1 . .5 4.2.2 . .6 4.3 Unitary representations . .6 5 Kirillov formula: the case of SU(2) 7 5.1 .....................................7 5.2 .....................................8 5.3 .....................................9 5.4 ..................................... 10 5.5 ..................................... 11 5.6 ..................................... 11 5.7 ..................................... 12 6 Kirillov formula: the general statement 12 7 The Heisenberg group 15 7.1 ..................................... 15 7.2 ..................................... 15 7.3 ..................................... 16 7.4 ..................................... 17 7.5 ..................................... 18 1 1 Overview These notes are recorded to complement the lectures. Material here has been rearranged from how it was presented in lecture to serve as a reference rather than an introduction. A principal aim of the lectures is that after attending them, you should be able to do the homework problems given below their summaries. 2 Lecture summaries and homework 2.1 2/21 and 2/22: Kirillov formula for SU(2) Objectives. You should be able to state the Kirillov formula for compact Lie groups and to prove it for SU(2). Summary. We proved the Kirillov formula for SU(2) by explicitly comput- ing the Fourier transforms of the characters of the irreducible representations. We philosophized about the possibility of a “combinatorial proof” and indicated the aims of the course. We stated the general Kirillov formula in the nilpotent and compact cases. By computing the canonical symplectic forms on the coad- joint orbits of SU(2), we verified that the general formula specialized to what we had shown directly. Homework 1 (Due Mar 8). 1. Skim at least the first few pages of Chapter 1 of Kirillov’s book (which should be available via mathscinet from the ETH network). The remaining exercises concern G := SU(2) and the representation πn : n n−1 n G ! GL(Vn), as in the lecture, with standard basis X ;X Y; : : : ; Y ∼ 3 ∼ ∗ and character χn. We fix coordinates g = R = g as in lecture. We ∗ denote by Oπn ⊆ g the coadjoint orbit of πn (i.e., the sphere of radius n+1) and fix a G-invariant inner product h; i on Vn, with associated norm kvk = hv; vi1=2. 1 2. Let φ1; φ2 2 Cc (R) with sin(θ) φ2(θ) = φ1(θ) for all θ 2 : θ R Write dθ for Lebesgue measure on the real line. For k 2 R, set φcj(k) = R ikθ e φj(θ) dθ. Let νj denote the compactly supported measure on g given θ R R in standard coordinates by g f dνj = θ f(0; 0; θ)φj(θ) dθ. (a) Using Kirillov’s formula and Archimedes’s theorem, verify that Z Z n+1 x p dk χn(e ) j(x) dν1(x) = φc1(k) ; x2g k=−n−1 2 where dk denotes Lebesgue measure. 2 R x (b) Define T 2 End(Vn) by T := x2g πn(e ) dν2(x). Verify that T is n+k n−k diagonalized by the basis X 2 Y 2 of πn, where k = −n; −n + 2; : : : ; n, with eigenvalues φc2(k), hence that Z x X χn(e ) dν2(x) = trace(T ) = φc2(k): x2g k=−n;−n+2;:::;n p (c) Verify that dν2 = j dν1. Conclude that Z n+1 dk X φc1(k) = φc2(k): 2 k=−n−1 k=−n;−n+2;:::;n [This may be deduced more directly by noting that the Fourier trans- form of sin(x)=x is a multiple of the characteristic function of the interval [−1; 1] and that the Fourier transform transports multiplica- tion to convolution. The point is to observe the relationship between this fact and the SU(2) case of the Kirillov formula.] 3. Following the steps outlined below, establish the following: there is a constant C > 0 so that for all n 2 Z≥0 and each ξ 2 Oπn , there is a unit vector v 2 πn so that for all x 2 g, ihx,ξi p 1=2 kπn(exp(x))v − e vk ≤ C( njxj + jxj ): (1) The moral is that the vector v is an approximate eigenvector, with eigenvaluep corresponding to ξ, of group elements g = exp(x) satisfying jxj < "= n for some small " > 0. ∗ (a) Using that Ad (G) acts transitively on Oπn , reduce to the case ξ = (0; 0; n + 1): [Hint: if (1) holds for a given pair (ξ; v), then a modified form of (1) ∗ holds for (Ad (g)ξ; πn(g)v).] (b) Choose v 2 πn to be a unit vector that is a multiple of the monomial Xn. Using the G-invariance of the inner product h; i, verify that v is orthogonal to the vectors Xn−1Y; : : : ; Y n and that a b hπ (g)v; vi = an g = 2 G: n for all c d (2) (c) Using the triangle inequality, reduce to the case p jxj ≤ 1= n: (3) ix ix + x For x = 3 2 1 2 g satisfying (3), write exp(x) = ix2 − x1 −ix3 a b c d , and verify that 2 ix3 2 a = 1 + ix3 + O(jxj ) = e (1 + O(jxj )) (4) 3 and an = einx3 (1 + O(njxj2)) = ei(n+1)x3 + O(njxj2 + jxj); (5) where O(A) denotes a quantity bounded in magnitude by CjAj for some constant C > 0. (d) Square both sides of (1), expand the norm squared as a sum of four inner products, and apply (2) and (5) to conclude. ∗ 4. For λ > 0, let Oλ := fξ 2 g : jξj = λg denote the corresponding coadjoint orbit and !λ its canonical symplectic volume measure, of total volume λ. ∗ ∗ ∗ Let p : g × g ! g denote the addition map. For λ1; λ2 > 0, verify the measure disintegration Z λ1+λ2 p∗(!λ1 ⊗ !λ2 ) = !λ dλ, jλ1−λ2j ∗ i.e., that for f 2 Cc(g ), Z Z Z λ1+λ2 Z f(x+y) d!λ1 (x) d!λ2 (y) = ( f(z) d!λ(z)) dλ, x2Oλ1 y2Oλ2 jλ1−λ2j z2Oλ where dλ denotes Lebesgue measure. Compare with the decomposition ∼ m+n πm ⊗ πn = ⊕k=jm−njπk: 4 3 Prerequisites • basics on Lie groups, e.g., definitions of Lie algebra, exponential map, adjoint action • basic language of representation theory (e.g., of finite groups) • definition of Haar measure; reading the Wikipedia article should suffice 4 Some background 4.1 Peter–Weyl theorem Let G be a group. Recall that f : G ! C is a class function if it is constant on conjugacy classes: f(ghg−1) = f(h). Assume henceforth that G is compact. Then it has a Haar measure dg, which we may normalize to have total volume one. The square-integrable class functions on G form a Hilbert space with inner product Z hf1; f2i := f1(g)f2(g) dg: g2G Let π : G ! GL(V ) be a finite-dimensional representation. Its character χ = χπ = χV : G ! C is the class function given by χ(g) := trace(π(g)). Theorem 1 (Peter–Weyl theorem). Every finite-dimensional representation of G decomposes as a finite direct sum of irreducible subrepresentations. The char- acters of the irreducible finite-dimensional representations of G, taken up to isomorphism, form an orthonormal basis for the space of class functions. In the special case that G is finite, this is proved in a first course on repre- sentation theory. The case of compact G may be proved similarly assuming the existence of Haar measure. We note also that for a finite-dimensional represen- tation V , V is irreducible iff hχV ; χV i = 1; as follows from (the easy part of) Theorem 1. If π : G ! GL(V ) is any finite-dimensional representation, then Maschke’s theorem (proved as in the case of finite groups by averaging with respect to Haar measure) shows that there is a G-invariant inner product on V , which is moreover unique if V is irreducible. 4.2 Symplectic manifolds 4.2.1 Let W be a vector space of finite-dimension over the reals. A symplectic form on W is a map σ : W × W ! R that is 5 1. bilinear, 2. alternating (σ(v; v) = 0, hence σ(u; v) = −σ(v; u)), and 3. non-degenerate: the map W ! W ∗ given by v 7! σ(v; ·) is an isomor- phism. The first two properties say that σ 2 Λ2(W )∗ =∼ Λ2(W ∗): Lemma 2. If W admits a symplectic form σ, then W is even-dimensional, say dim(W ) = 2n, and there is a basis e1; : : : ; en; f1; : : : ; fn of W so that σ(ei; ej) = 0 = σ(fi; fj) and σ(ei; fj) = δij = −σ(fi; ej). 0 Proof sketch. Choose any nonzero e1, choose f1 with he1; f1i = 1, set W := fv 2 W : hv; e1i = hv; f1i = 0g, and continue. ∗ ∗ ∗ Denoting by e1; : : : ; fn the basis of W dual to that furnished by the lemma, we have ∗ ∗ ∗ ∗ σ = e1 ^ f1 + ··· + en ^ fn: We check readily that σn := σ ^ · · · ^ σ = n!e∗ ^ f ∗ ^ · · · ^ e∗ ^ f ∗ 2 Λ2n(W ∗): | {z } 1 1 n n n times In particular, σn 6= 0. 4.2.2 Let M be a manifold. Recall that a 2-form σ on M is the smooth assignment 2 ∗ M 3 ξ 7! σξ 2 Λ (Tξ M), thus σξ is a bilinear alternating pairing on the tangent space TξM. Its exterior derivative dσ is a 3-form; recall that σ is closed if dσ = 0.