Weights, roots and the Weyl integral formula

Weights, roots and the Weyl integral formula

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 10 April 2010

Weights and roots

Let G be a compact connected group. A maximal torus of G is a maximal connected subgroup of G isomorphic to S 1 k for some positive integer k.

Fix a maximal torus T in G. The group T is a maximal connected abelian subgroup of G. The Weyl group is W= N G T /T,where N G T = gG| gT g -1 =T . The Weyl group W acts on T by conjugation. The map G/T×T φ G gT t gt g -1 is surjective and Card φ -1 g = W for any gG. It follows from this that

  1. Every element gG is in some maximal torus.
  2. Any two maximal tori in G are conjugate.
Thus, maximal tori, are unique up to conjugacy and cover the group G.

Let P be an index set for the irreducible representations of T. Since the irreducible representations of S 1 are indexed by ,P k . The set P is called the weight lattice of G. If λPthen X λ:T* , denotes the corresponding irreducible representation of T. The W -action on T induces a W -action on P via X wλ t = X λ w -1 t ,for alltT. A representation V of G is a representation of T, by restriction, and, as a T -module, V= λP V λ ,where V λ = vV| tv= X λ t v  for all  tT.

The vector space V λ is the X λ isotypic component of the T -module V. The W -action on T gives dim V λ =dim V wλ ,for allwW  and  λP. The vector space V λ is the λ -weight space of V. A weight vector of weight λ in V is a vector v in V λ .

Let G be a compact connected Lie group and let 𝔲=Lie G . The group G acts on 𝔲 the adjoint representation. Extend the adjoint representation of G on the complex vector space 𝔤 =𝔲i𝔲=𝔲. By ??? this representation extends to a representation of the complex algebraic group G which is the complexification of G. Since G is compact, the adjoint representation of G of 𝔤 , and thus the adjoint representation of 𝔤 on itself, is completely decomposable. This shows that 𝔤 is a complex semisimple Lie algebra.

The adjoint representation 𝔤 of G has a weight decomposition 𝔤 = αP 𝔤 α , and the root system of G is the set R= αP| α0, 𝔤 α 0 of nonzero weights of the adjoint representation. The roots are the elements of R. Set 𝔥= 𝔤 0 . Then 𝔤 =𝔥 αR 𝔤 α is the decomposition of 𝔤 into the Cartan subalgebra 𝔥 and root spaces 𝔤 α . (Note that the usual denotation is 𝔥 =i𝔥, 𝔥 =𝔥i𝔥, where 𝔥 is a Cartan subalgebra of 𝔤, ie a maximal abelian subspace of 𝔤. Also 𝔤 0 . Also 𝔤 0 = 𝔥 since 𝔥 is maximal abelian in 𝔤. Also 𝔥=𝔱i𝔱 where 𝔱 is the Lie algebra of the maximal torus T of G, and the maximal abelian sublgebra in 𝔤. Don't forget to think of X: T * t X λ t e h e λ h and λ: 𝔥 h λ h

The Weyl group W is generated by s α ,αR. The action of W on 𝔥* is generated by the transformations s α : 𝔥* 𝔥* λ λ- λ α where α = 2α αα , and , :𝔥*×𝔥* is a nondegenerate symmetric bilinear form.

  1. If α is a root then -α is a root and these are the only two multiples of α which are roots. (The thing that makes this work is that the root spaces are purely imaginary).
  2. If α is a root the dim 𝔤 α =1.
  3. The only connected compact Lie groups with dim T =1 are S O 3 and the twofold simply connected cover of S O 3 .

Proof 1) Suppose that α is a root and that x 𝔤 α . X α : T * e h e α h and X -α : T * e h e α h = e -α h since α h i for h𝔱. Then, for all h𝔱, h x - = h- x- = hx = α h x - =-α h x - , and so x - 𝔤 -α . Thus 𝔤 -α 0 and -α is a root. Note that x x - 𝔥 since it has weight 0.

2) Consider X α :T * . Then T α =ker X α is closed in T and is of codimension 1. Let T α be the connected component of the identity in T α and let Z α = Z G T α be the centraliser of T α in U (this is connected.) Then Lie Z α =𝔱i𝔱 h T α ,β h =1 𝔤 β =𝔥 k 𝔤 kα.

Now Z α Z α / T α | | T T/ T α So T/ T α is a maximal torus of Z α / T α and dimT/ T α =1. Then Lie Z α = 𝔥 α H α k 𝔤 kα .

If X α 𝔤 α then X α X -α =λ H α and λ0 since H is maximal abelian in Lie Z α / T α = k 𝔤 kα . Now consider the action of H α on H k >0 𝔤 kα X α . Then Tr H = 1 λ Tr X α X -α = 1 λ ad X α ad X -α - ad X -α ad X α =0. But this implies 0=0+ k >0 dim 𝔤 kα kα H α -α H α . So 𝔤 kα =0 for k>1 and 𝔤 α = X α . So span X α X -α H α is a three dimensional subalgebra of 𝔤.

If U is a compact connected Lie group such that dimT=1 then U has Lie algebra 𝔤= span X α X -α H α =𝔲i𝔲. Then the Weyl group of U is 1 s α S 2 where s α comes from conjugation by an element of Z α and so s α leaves T α fixed.

So the Weyl group of G contains all the s α ,αR.

Example There are only two compact connected groups of dimension 3, SO 3 andSpin 3 .

Proof G acts on 𝔤 and this gives an imbedding Ad:GSO 𝔤 (with respect to an Ad invariant form on 𝔤. ) This is an immersion since everything is connected. So G is a conver of SO 3 .

Weyl's integral formula

Let G be the compact connected Lie group. Let T be a maximal torus of G and let W be the Weyl group. Let R be the set of roots. Then W G f x dx= T αR X α t -1 G f gt g -1 dgdt.

Proof First note that the map G/T×TG given by gT t gt, can be used to define a (left) G invariant measure on G/T so that G f g dg= G/T×T f gt dtd gT , and thus, for yT, G f gy g -1 dg = G/T×T f gty t -1 g -1 dtd gT = G/T f gy g -1 dtd gT = G/T×T f gy g -1 d gT .

Then the map φ:G/T×TG given by gT t gt g -1 yields

W G f g dg= G/T×T f gt g -1 J gT t dtd gT ,
where J gT t os the determinant of the differential at gT t of the map φ. By translation, J gT t is the same as the determinant of the differential at the identity, Te , of the map L g t -1 g -1 φ L gt , G/T×T G/T×T G G xT y gxT ty gx ty gx -1 g t -1 g -1 gx ty gx -1 . Since g t -1 g -1 gx ty gx -1 =g t -1 xty x -1 g -1 this differential is 𝔤/𝔥 𝔤 XY Ad g Ad t -1 X +Y-X . So J gT t is the determinant of the linear transformation of 𝔤 given by Ad 𝔤 g Ad 𝔤/𝔥 t -1 - id 𝔤/𝔥 0 0 id 𝔥 , where the second factor is a block 2×2 matrix with respect to the decomposition 𝔤/𝔥𝔥 and Ad 𝔤/𝔥 is the adjoint action of T restricted to the subspace 𝔤/𝔥 in 𝔤. The element t -1 acts on the root space 𝔤 α by the value X α t -1 where X α :T* is the character of T associated to the root α. Since G is unimodular, det Ad g =1, and since 𝔤/𝔥 = αR 𝔤 α ,
J gT t = αR X α t -1 -1 = αR X α t -1 ,
where the last equality follows from the fact that if α is a root then -α is also a root. Then combine the numbered equations above to prove the theorem.

It follows from this theorem that, if χ and η are class functions on G then

G χ g η g dg = 1 W T αR X α t -1 G χ gt g -1 η gt g -1 dgdt = 1 W T α>0 X α t -1 X -α t -1 G χ t η t dgdt = 1 W T α>0 X α 2 t - X - α 2 t X - α 2 t - X α 2 t G χ t η t dgdt = 1 W T α>0 a ρ η t dt.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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