WEYL INTEGRATION FORMULA

Transcription

1 WEYL INERAION FORMULA YIFAN WU, Abstract. In this exposition, we will prove the Weyl integration formula, which states that if is a compact connected Lie group with maximal torus, and if f L then fgdg = fgtg dg detid g Adt g/t dt. W he contents are based on modified from lectures given by Upamanyu Sharma. Contents. Review of Weyl s Covering heorem and Integration on Lie roups 2. Relation between w, w, and w 3 3. Weyl Integration Formula for Continuous Functions 4 4. Weyl Integration Formula for L Functions 7 References 7. Review of Weyl s Covering heorem and Integration on Lie roups hroughout this exposition, we denote to be a compact conected Lie group with Lie algebra g. Let be a maximal torus of. he Lie algebra of is denoted t, which is a Cartan subalgebra of g. he complexifications g R C and t R C of g and t are denoted g C and t C, respectively. g C admits a Cartan decomposition g C = t C α Φ g α, where Φ is the collection of non-zero roots α t C and g α = {x g C : αhx = [h, x] for all h t C }. Date: First Version: February 24, 208. Last Update: March 9, 208.

2 he real Lie algebra g is decomposed as Weyl Integration Formula g = t g α Φ g α. he following two lemmas are the key ingredient to prove the Weyl s theorem.3. Lemma.. Let π : be the natural projection. here exists an open neighborhood U g of 0 in g α Φ g α such that he following hold: Let U = exp U g and U = πu. hen the composition exp U g U π U is a diffeomorphism. 2U is an open neighborhood of e in. 3Let U = {gt : g U, t }. hen U is an open neighborhood of e in. 4Let ξ : U, ξgt = g, t. hen ξ is a well-defined diffeomorphism onto U. Lemma.2. Let U be as in the previous lemma.. Let g, t. Consider φ : U given by the composition U ξ U L g L t hen the differential satisfies the following: Let h t and x g α Φ g α. hen ψ L gt g. dφ e : g g, dφ e h + x = AdgAdt Id g x + h. 2he determinant is given by detdφ e = dt = α Φ ξ α t = detid g Adt g/t. Weyl s covering theorem itself says the following: heorem.3. Let W = N / be the Weyl group of. he map ψ : reg reg, ψg, t = gtg is a smooth surjective W to local diffeomorphism. We now recall some basic results regarding integration on Lie groups. Up to ±, there exist unique left invariant differential forms of top degree w top, w top, and w top such that w =, w =, w =. 2

3 Yifan Wu, Using the Riesz representation theorem, there exist unique left and right invariant Borel measures dg on, dt on, and dg on such that F gdg = F w, F 2 tdt = F w, F 3 g dg = F 3 w for F C, F 2 C, and F 3 C. Let π : denote the natural projection map. hen we have fg dg = F πgdg /H for all f L, and F gdg = for all F L. F gtdtdg 2. Relation between w, w, and w Let w, w, and w be as in the previous section. Let n = dim and l = dim. hen w is a differential n form on, w is a differential l form on, and w is a differential n l form on. he pullback π w is a differential n l form on. In this section, we will see there is a differential l form w on such that π w w = w on. We will also see that w can be written as a pullback of w and w. he precise statement is recorded in the following proposition. he proof is very tedious and will only be sketched here. Full details can be found in [4, Lemma 7.3, page 59]. Proposition 2.. Let ι : denote the closed embedding, and π :, π 2 : be the two projection maps. Recall from lemma. that ξ : U, ξgt = g, t is a well defined diffeorphism onto U, where U is an open neighborhood of e in. here exists a left invariant form differential l form w l on such that the following hold: As differential forms on, w = ι w. 2As differential forms on U, π w = ξ π w. 3here exists some differential l form w l U such that on U, ξ π 2w = w + w. 3

4 Weyl Integration Formula 4On U, 5On, w = ξ π w π 2w. w = π w w. Proof Sketch. he restriction ι e : g t is surjective. Hence there exists an alternating l form w e l g such that ι e w e = w e l t. Using left translations, w e is uniquely extended to a left invariant differential l form w l on. he wedge product π w w is a differential n form on. A short calculation shows that π w w is left invariant on. Up to scalar, w is the unique left invariant differential n form on. herefore, π w w = cw for some c R. he remaining of proof will be devoted to showing c = and will be omitted. 3. Weyl Integration Formula for Continuous Functions We will use the result from the previous section and the Weyl s covering theorem to prove the following proposition. Recall from theorem.3 that ψ : reg reg, ψg, t = gtg is a smooth surjective W to local diffeomorphism. Proposition 3.. here exists a smooth function δ : R such that 3.2 ψ w gtg = δπw π2w for all g, t. 2We have where dt is given as in lemma.2: ψ w = dtπ w π 2w g,t dt = α Φ ξ α t = detid g Adt g/t Proof. Note ψ w and π w π 2w are both top degree differential forms on. hen on each point g, t, they differ by a mutiple of a constant δg, t. 4

5 Yifan Wu, 2Recall from part 2 and part 4 of lemma. that U is an open neighborhood of e in and ξ : U, ξgt = g, t. is a well-defined diffeomorphism onto U. By part 4 of proposition 2., w = ξ π w π 2w on U. hen on the open neighborhood U of e, e in, we have In particular at e, e, 3.3 ξ w e = ξ w = π w π 2w. πw π2w. e,e By left invariance of w, we have L gt g w e = w gtg. herefore, ψ L gt g w e = ψ w gtg 3.2 = L g L t δπ w π2 w e,e 3.3 = L g L t ξ δ L g L t ξw As in lemma.2, φ : U is given by the composition e. hen U ξ U L g L t ψ L gt g. φ w e = L gt g ψ L g L t ξ w e = ξ L g L t ψ L g g w e = ξ L g L t L g L t ξ δ L g L t ξw = δ L g L t ξw herefore, detdφ e = δ L g L t ξe = δg, t. he determinant detdφ e is precisely dt, as in lemma.2. Part implies the desired result holds. Using this proposition, we can deduce Weyl integration formula for continuous functions. Before that, recall both \ reg and \ reg have measure zero. 5 e. e

6 heorem 3.4. Let f C. hen fgdg = W Proof. Since = Weyl Integration Formula fgtg dg dtdt fgtg dg detid g Adt g/t dt. W ψ : reg reg, ψg, t = gtg is a smooth surjective W to local diffeomorphism, we have fgdg = fw = ψ fw = reg W reg ψ reg W by change of variable formula. By part 2 of proposition 3., we have It follows that by Fubini s theorem, f ψψ w = = reg reg ψ w = dtπ w π 2w. reg f ψw dtw = Combining these, we get = reg fgdg = reg reg reg f ψdtπ w π 2w f ψg, tdg dtdt fgtg dg dtdt. fgtg dg dtdt. W reg Since both \ reg and \ reg have measure zero, we have fgdg = fgtg dg dtdt, W f ψψ w as desired. 6

7 Yifan Wu, 4. Weyl Integration Formula for L Functions he previous section settles down the Weyl integration formula for all f C. In fact, the Weyl integration formula holds for all f L. In this section, we will state the result but omit the proof. he full details for the proof can be found in [, heorem 3.4., page 85]. Lemma 4.. W acts on L via for F L, n N. n F g, t = F gn, ntn heorem 4.2. Let L W denote the W invariant integrable functions on. Consider the mapping C C that maps f C to F C where F is the integrand in theorem 3.4: F g, t = fgtg dt. his mapping extends to a topological isomorphism L L W. 2If f L, then fgdg = fgtg dg dtdt. W References [] J. J. Duistermaat, J. A. C. Kolk, Lie roups, Springer-Verlag, Berlin Heidelberg, [2] J. Humphreys, Introduction to Lie Algebras and Representation heory, Springer-Verlag, New York, 972. [3] J. Lee, Introduction to Smooth Manifolds, Springer, New York, 203. [4] M. R. Sepanski, Compact Lie roups, Springer, New York,