The Fine Structure of Translation Functors

Home , Translation functor

REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 3, Pages 223–249 (August 16, 1999) S 1088-4165(99)00056-4

THE FINE STRUCTURE OF TRANSLATION FUNCTORS

KAREN GUNZL¨

Abstract. Let E be a simple ﬁnite-dimensional representation of a semisimple Lie algebra with extremal weight ν and let 0 = e Eν .LetM(τ)bethe 6 ∈ Verma module with highest weight τ and 0 = vτ M(τ)τ . We investigate the 6 ∈ projection of e vτ E M(τ) on the central character χ(τ + ν). This is a ⊗ ∈ ⊗ rational function in τ and we calculate its poles and zeros. We then apply this result in order to compare translation functors.

Contents 1. Introduction 2. Filtration of E M(τ) 3. The fine structure⊗ of E M(τ) ⊗ 3.1. Algebraicity of fν 3.2. Poles of fν. Theorem 1 and proof 3.3. Proof of Lemmas 6, 7, 8, and 9 4. The triangle function ∆ 4.1. Preliminaries 4.2. Definition of ∆ 4.3. Uncanonical definition of ∆ 5. Bernstein’s relative trace and a special case of ∆ 5.1. The relative trace 5.2. The special case ∆( ν, ν; w0) 5.3. Proof of Theorem 2− τ τ+ν 5.4. The case M(τ) Tτ+νTτ M(τ) M(τ) 6. Calculation of ∆ → → 7. Outlook 7.1. Identities 7.2. Generalizations References

1. Introduction Let k be a ﬁeld of characteristic zero and let g be a split semisimple Lie algebra over k.Thenh b gdenote a Cartan and a Borel subalgebra, U = U(g)the ⊂ ⊂ + universal enveloping algebra and Z U its center. We set R R h∗ to be the ⊂ ⊂ ⊂ roots of b and of g.Letn(resp. n−)bethesumofthepositive(negative)weight

Received by the editors September 2, 1998 and, in revised form, July 19, 1999. 1991 Mathematics Subject Classiﬁcation. Primary 17B10. Partially supported by EEC TMR-Network ERB FMRX-CT97-0100.

c 1999 American Mathematical Society

223 224 KAREN GUNZL¨

+ spaces and denote by P P h∗ the set of dominant and the lattice of integral weights respectively. Let ⊂ be⊂ the Weyl group. W A g-module M is called Z-finite,ifdimk(Zm)< for all m M.EveryZ- finite module M splits under the operation of the center∞ Z into∈ a direct sum of submodules M = χ MaxZ Mχ.Hereχruns over the maximal ideals in the ∈ center and M ∞(χ), where χ ∈ML n ∞(χ):= M for all m M there exists n N such that χ m =0 . M { | ∈ ∈ } We denote the projection onto the central character χ by prχ : M Mχ.By means of the Harish-Chandra homomorphism [Di, 7.4] ξ : Z S(h) (normalized7→ → by z ξ(z) Un z Z) we assign to each weight τ h∗ its central character χ(τ) − ∈ ∀ ∈ ∈ defined by χτ (z):= ξ(z) (τ) z Z. Let now ν P be an integral∀ ∈ weight and denote by E := E(ν) the irreducible finite-dimensional∈ g-module with highest weight in ν. It is known that for each Z-finite module M the tensor product E M is againW Z-finite (see [Ko]). If we ⊗ now tensor a module M ∞ χ(τ) with E and then project onto the central character χ(τ + ν), we obtain∈M the translation functor τ +ν T : ∞ χ(τ) ∞ χ ( τ + ν ) τ M −→ M M pr (E M). 7→ χ(τ+ν) ⊗ In the present paper we investigate the fine structure of translation functors. Namely, take for M the Verma module M(τ):=U(g) k with its highest ⊗U(b) τ weight vector v := 1 1 M(τ) and then identify for all τ h∗: τ ⊗ ∈ τ ∈

M(τ) ∼ U ( n − ) uv −→ u. τ 7→ We choose a fixed extremal weight vector e E and define the map ν ∈ ν fν : h∗ E U ( n − ) τ −→ pr ⊗ (e v ). 7→ χ(τ+ν) ν ⊗ τ Here we identify pr (e v ) E M(τ) with its image in E U(n−). The χ(τ+ν) ν ⊗ τ ∈ ⊗ ⊗ image of the map fν is then contained in the finite-dimensional ν-weight space of E U(n−), and we may thus regard f as a map between varieties. In general ⊗ ν however, fν is not a morphism. Let ρ := 1/2 α R+ α denote the half sum of positive roots, α∨ the co-root of α and set ∈ P + ν := (α, mα) R Z ρ, α∨ mα < ν + ρ, α∨ . N { ∈ × |−h i≤ −h i} Then the set

:= τ h∗ τ,α∨ = m Hν { ∈ |h i } (α,m) [∈Nν is a finite family of hyperplanes and therefore Zariski closed. We will show at first that fν is a morphism of varieties on the complement of ν ,where h∗ is a suitable Zariski closed subset of codimension 2. MoreH precisely,∪S consistsS⊂ of intersections of finitely many hyperplanes. ≥ S THE FINE STRUCTURE OF TRANSLATION FUNCTORS 225

Deﬁne for all roots α R and for all m Z the polynomial function Hα,m on ∈ ∈ h∗ by

H (τ):= τ,α∨ mfor all τ h∗. α,m h i− ∈ If we then set δν := Hα,m,weobtain ν = τ h∗ δν(τ)=0.A (α,m) ν H { ∈ | } central result of this paper will∈N be the following Q Theorem. There exists a morphism of varieties G : h∗ (E U(n−))ν ,such that the set of zeros of G has codimension 2 and such→ that G⊗equals δ f on ≥ ν ν h∗ ( ). − Hν ∪S This means that the map fν has a pole of order 1 along each of the hyperplanes τ,α∨ = m with (α, m) ν and outside of these hyperplanes it is a non-vanishing hmorphismi of varieties except∈N on a set of codimension 2. ≥ We remark that Kashiwara [Ka, Thm. 1.7] has also determined these poles (for universal Verma modules) in the special case of dominant τ and antidominant ν, i.e. for ν the lowest weight of E. He then used this result to calculate b-functions on the flag manifold. Etingof and Styrkas [ES] have considered a slightly different situation in con- nection with intertwiners: Instead of looking at the element pr (e v ) χ(τ+ν) ν ⊗ τ which for generic τ is a generator for the Verma module M(τ + ν) seen as a submodule of E M(τ), consider rather a different generator w + given by ⊗ τ ν wτ+ν := eν vτ + R,whereR E U(n−)n−vτ. By the above theorem we ⊗ ∈ ⊗ 1 can then express pr (e v )as(δ(τ))− a (τ)w + ,wherea is a polyno- χ(τ+ν) ν ⊗ τ ν ν τ ν ν mial in h∗ and its zeros contain the poles of the coefficient functions occurring in R.Forν= 0, not necessarily an extremal weight of E, these poles (i.e. the poles of wτ+ν ) have been computed in [ES] in terms of the determinant of the Shapovalov form. Etingof and Varchenko later used this result to establish a Hopf-algebroid structure on a certain (rational) exchange quantum group [EV, Ch. 5].

In Chapter 4 we introduce the so-called triangle functions ∆(µ, ν; x)(τ) for integral weights µ and ν and x . These are rational functions on h∗, which measure ∈W τ+ν+µ τ+ν in a subtle way the relation between the two translation functors Tτ+ν Tτ τ+ν+µ ◦ and Tτ by ﬁrst applying them to the Verma modules M(τ)andM(x τ)and then identifying the results with M(τ + ν + µ)andM(x (τ+ν+µ)) respectively.· This construction is the characteristic zero analogue· of a function introduced by Andersen, Jantzen and Soergel [AJS, Ch. 11]. In fact, the motivation for the present work was to understand better some of the more technical chapters in [AJS] and in particular realizing the poles and zeros of ∆ in a diﬀerent way. In order to do this we make use of the maps fν for suitable integral weights ν and can thus write ∆ as a product of the corresponding δν’s. This is done in Chapter 6 (Lemma 11). If we set nowα ¯(λ):=1for λ, α∨ < 0and¯α(λ):=0for λ, α∨ 0, we then obtain h i h i≥ Theorem. Let ν, µ P be integral weights and x .Then ∈ ∈W τ,α α¯(ν) τ + ν, α α¯(µ) τ + ν + µ, α α¯(ν+µ) ∆(µ, ν; x)(τ ρ)=c ∨ ∨ ∨ h iα¯(ν+hµ) i α¯(hν) i α¯(µ) − τ,α∨ τ + ν, α∨ τ + ν + µ, α∨ α R+ ∈ h i h i h i with Yxα R ∈ − for a constant c k× independent of τ,ν,µ and x. ∈ 226 KAREN GUNZL¨

Bernstein deﬁned in [Be] the so-called relative trace trE for a ﬁnite-dimensional vector space E and this function is related to the special case ∆( ν, ν; w0), where − ν is dominant and w0 is the longest element of . This is explained in Chapter W 5. By Bernstein’s explicit formula for trE – or alternatively by [Ka, Thm. 1.9]—we obtain

+ Corollary. Let ν P be dominant and τ h∗ generic, i.e. τ,α∨ / Z for all α R.Then ∈ ∈ h i ∈ ∈ τ+ν+ρ, α∨ ∆( ν, ν; w0)(τ)= h i . − τ + ρ, α∨ α R+ Y∈ h i 2. Filtration of E M(τ) ⊗ In the following let ν P be a ﬁxed integral weight. Then let ν1,... ,ν be the ∈ n multiset of weights of E = E(ν), i.e. with multiplicities, and let (ei)1 i n be a basis of weight vectors of E, such that e E and ν <ν id1+d2+ dj 1 X X··· − We obtain immediately: The chain E M(τ)=M1 M 0isa ⊗ ⊃ ··· ⊃ m ⊃ ﬁltration of submodules such that Mj/Mj+1 = M(τ + µj) and the summands ∼ dj M(τ + µ ) are generated by the vectors (e v )/M +1 =pr(e v )/M +1,where j i ⊗ τL j i⊗ τ j ei is a vector of weight µj.

Lemma 1. For 1 j m let sj Z with χτ +µj (sj )=0and set zi := j i sj . ≤ ≤ ∈ ≥ Then for 1 i m we get: ziMi =0. In particular,≤ ≤ for a weight vector e E we obtain Q µi ∈ µi z +1(e v )=z+1 pr (e v ). i µi ⊗ τ i χ(τ+µi) µi ⊗ τ Proof. The ﬁrst statement follows by induction from above and since a central element z Z operates on the Verma module M(λ) by multiplication with the ∈ scalar χλ(z). By construction of the M it is clear, that for e v M its projections i µi ⊗ τ ∈ i prχ(τ+µ)(eµi vτ ) lie already in Mi+1 if χ(τ + µ) = χ(τ + µi). But eµi vτ equals the sum of its⊗ projections and thus the second statement6 follows from the⊗ ﬁrst.

Note that for the extremal weight ν = νi0 = µj0 its weight space is of dimension 1. Therefore, we can always choose the numbering of the ν such that i>i0 i ⇐⇒ ν >ν =νor equivalently j>j0 µ >µ =ν. It then follows immediately i i0 ⇐⇒ j j0 that an element of Mi0 = Mj0 (of Mi0+1 = Mj0+1 resp.) consists only of parts with THE FINE STRUCTURE OF TRANSLATION FUNCTORS 227

central character χτ+µ for µ P (E) with µ ν (µ>νresp.). For eν vτ Mi0 we thus obtain the following special∈ case of Lemma≥ 1. ⊗ ∈

Lemma 2. For µ P (E ) with µ>νlet sµ Z such that χτ +µ(sµ)=0and set z := s .Then∈ z(e v)=zpr (e∈ v ). µ>ν µ ν⊗ τ χ(τ+ν) ν ⊗ τ Q 3. The fine structure of E M(τ) ⊗ 3.1. Algebraicity of f . For ν P fixed, set E := E(ν) and let f be the map ν ∈ ν fν : h∗ E U ( n − ) τ −→ pr ⊗ (e v ) 7→ χ(τ+ν) ν ⊗ τ whereweidentifyE U(n−)=E M(τ) for all τ h∗. We will first construct a ⊗ ∼ ⊗ ∈ Zariski open set h∗ such that f restricted to is a morphism of varieties. In U⊂ ν U this case we will also say that fν is algebraic on . Before we go on, we need some more notation:U The dot-operation of the Weyl group on h∗ has fixed point ρ = 1/2 α R+ α andisdefinedbyw λ:= − − ∈ · w(λ + ρ) ρ.Let (h∗) denote the set of polynomial functions on h∗ and (h∗)W· the ( )-invariants.− P We define the operatorP P W· sym : (h∗) ( h ∗ ) W· P −→ P by (sym s)(λ):= x s(x λ) for all λ h∗,s (h∗). ∈W · ∈ ∈P For any µ P (E ) with µ>νwe now choose an element hµ h, hµ =0and ∈ Q ∈ 6 define maps H : h∗ h∗ k by µ × −→ H (λ, τ):= λ τ µ, h for all λ, τ h∗. µ h − − µi ∈ If we fix τ h∗,themapH ( ,τ): h∗ k is then a polynomial function on ∈ µ − −→ h∗ and its kernel is obviously a hyperplane in h∗. We then define a ( )-invariant W· polynomial function pτ , depending also on the choice of the elements hµ,by p := sym H ( ,τ). τ µ − µ P (E),µ>ν ∈ Y

By means of the Harish-Chandra isomorphism [Di, 7.4] ξ : Z ∼ ( h ∗ ) W·,there exist central elements s Z, such that ξ(s ) = sym H ( ,τ).−→ For Pz := s µ ∈ µ µ − τ µ>ν µ we have ξ(z )=p . Deﬁne now the map τ τ Q

u = u hµ : h∗ k { } τ −→ p (τ + ν) 7→ τ and set h := τ h∗ u(τ) =0 h∗. U{ µ} { ∈ | 6 }⊂

Lemma 3. The map fν restricted to hµ is a morphism of varieties and for all 1 U{ } τ h we have fν (τ)=(u(τ))− zτ (eν vτ ). ∈U{ µ} ⊗ Proof. We know χτ +µ (sµ )=(ξ(sµ))(τ + µ) = (sym Hµ( ,τ))(τ + µ)=0andby Lemma2weconcludez (e v )=z pr (e v ).− τ ν⊗ τ τ χ(τ+ν) ν ⊗ τ Let now τ h , i.e. u(τ) = 0. This means, that for all w and for all ∈U{ µ} 6 ∈W µ P (E) with µ>νwe have w (τ + ν) τ µ, hµ = 0 and in particular w ∈(τ + ν) = τ + µ or equivalentlyh χ· (τ + ν) =− χ(−τ + µ) i6µ P (E),µ > ν.This implies,· that6 already the projection pr (e 6 v ) E ∀M(∈τ) generates a Verma χ(τ+ν) ν ⊗ τ ∈ ⊗ module M(τ + ν). A central element z operates on this by multiplication with the 228 KAREN GUNZL¨ scalar (ξ(z))(τ + ν)=u(τ). We therefore get z (e v )=z pr (e v )= τ ν ⊗ τ τ χ(τ+ν) ν ⊗ τ u(τ)fν(τ) and the above equation follows. We have yet to show that fν is algebraic on h . By construction of zτ it U{ µ} is clear that z depends algebraically on τ for all τ h∗. Also the map τ τ ∈ 7→ z (e v ) E U(n−) is a morphism on h∗ and 1/u is per deﬁnition algebraic τ ν ⊗ τ ∈ ⊗ on h .Thusfν restricted to h is a morphism. U{ µ} U{ µ} Example. Let ν P (E) be an extremal and dominant weight. For all µ P (E) ∈ ∈ with µ = ν we then have µ<νand we may choose z := 1 Z for all τ h∗. 6 τ ∈ ∈ The above lemma then implies hµ = h∗,themapfν is a morphism on h∗ and f (τ)=e v for all τ. U{ } ν ν ⊗ τ Lemma 3 implies that fν is algebraic on := h , where the union is taken U U{ µ} over all possible choices for hµ µ P (E),µ>ν .Ifweset { | ∈ S} := τ h∗ u(τ)=p (τ+ν) = 0 for all choices of h , A { ∈ | τ { µ}} we can write as = h∗ .Since U U −A p(τ+ν)= w (τ + ν) τ µ, h τ h · − − µi µ P(E),µ>ν w ∈ Y Y∈W we know that τ is in if and only if there exists a w and a µ P (E) with µ>ν, such that w (τA+ ν)=τ+µ.Thusweobtain ∈W ∈ · = τ h∗ w (τ + ν)=τ+µ . A µ P(E),µ>ν { ∈ | · } ∈ w [∈W Let us examine more closely the sets on the right hand side: For w = s ,α R,we α ∈ get τ h∗ τ,α∨ α = ν µ ν+ρ, α∨ α , which implies that this set is nonempty { ∈ |h i − −h i } if and only if there exists a weight µ = ν τ+ν+ρ, α∨ α in the α-string through ν, such that µ P (E)andµ>ν.Since−hνwas an extremali weight, it is either the greatest or the∈ smallest weight in this α-string and hence such a µ exists only if + ν, α∨ < 0 for an α R .Inthiscaseallµ := ν + nα for 1 n ν, α∨ h i ∈ (n) ≤ ≤−h i are weights of E with µ ν. Comparing this with µ = ν ν+ρ+τ,α∨ α we (n) ≥ −h i obtain as a condition for τ, precisely ρ, α∨ τ,α∨ ν+ρ, α∨ 1 for an α R+. −h i≤h i≤−h i− ∈Set 0 := w w=s for an α R .For W { ∈W| 6 α ∈ } + ν:= (α, mα) R Z ρ, α∨ mα < ν + ρ, α∨ , N { ∈ × |−h i≤ −h i} := τ h∗ τ,α∨ = m , Hν { ∈ |h i } (α,m) [∈Nν := τ h∗ w (τ + ν)=τ+µ , S µ P (E),µ>ν { ∈ | · } ∈ w 0 [∈W we get = . Remark that in general and are not disjoint. We claim A Hν ∪S Hν S that codim 2: for w and µ P let (w, µ):= τ h∗ w (τ+ν)=τ+µ . Since is aS≥ ﬁnite union of∈W such sets it∈ will beS enough to show{ ∈ that| codim· (w, µ) }2 for w S 0 and µ>ν. First rewrite w (τ + ν)=τ+µas (w id)(τ)=S µ w≥ ν ∈W · − − · and note that codim (w, µ) = rank(w id) = dim h∗ mw,wheremw denotes the multiplicity of theS eigenvalue 1 of w −since w is diagonizable.− If we assume now codim (w, µ) 1, then m =dimh∗ or m =dimh∗ 1 which implies either S ≤ w w − THE FINE STRUCTURE OF TRANSLATION FUNCTORS 229

0 w = e or w = sα for some α R.Butforw and µ>νthis is impossible and we get in this case that codim∈ (w, µ) = rank(∈Ww id) 2. This implies also that a set (w,S µ) appearing in −consists≥ of an intersection of S S at least two integral hyperplanes in h∗.

UsingLemma3weobtain Lemma 4. The map f is algebraic on the complement of . ν Hν ∪S 3.2. Poles of fν . Theorem 1 and proof. Let now δν (h∗) be the product of the equations of the hyperplanes in ;thatis ∈P Hν δ (τ):= H (τ)= ( τ + ρ, α∨ n ), ν α,m h i− α (α,m) α R+ 0 n < ν,α Y∈Nν Y∈ ≤ αY−h ∨i where H (τ):= τ,α∨ m.Wethenhave = τ h∗ δ(τ)=0 . α,m h i− Hν { ∈ | ν } Theorem 1. There exists a morphism of varieties G : h∗ ( E ( ν ) U ( n − ))ν , such that the set of zeros of G has codimension 2 and such−→ that G equals⊗ δ f ≥ ν ν on h∗ ( ). − Hν ∪S The proof comes in two parts. In the ﬁrst part we demonstrate the existence of an algebraic extension G of δνfν on the whole of h∗, in the second we will show that the set of zeros of G has codimension 2. It will be useful to introduce the ≥ set 1 h∗ of all intersections of hyperplanes in .Soifweset S ⊂ Hν 1:= τ h∗ H (τ)=0=H (τ)for(α, m) =(β,n) , S { ∈ | α,m β,n 6 ∈Nν} then 1 is a Zariski closed subset of codimension 2. By deﬁnition of and S∪S ≥ S 1 we obtain immediately S + Lemma 5. Let τ ( 1). Then there exists exactly one α R , such that ∈Hν− S∪S ∈ s (τ+ν)=τ+µ0 for a weight µ0 of E with µ0 >ν, namely µ0 = ν τ+ν+ρ, α∨ α. α· −h i For all µ P (E ) with µ>νand µ = µ0 we have w (τ + ν) = τ + µ for all w . ∈ 6 · 6 ∈W Proof of Theorem 1. Again let = h∗ ( ν )bethesetonwhichfν is algebraic. First we claim U − H ∪S

( ) For all τ0 h∗ ( 1) there exists a Zariski open neighborhood 0 of τ0 ∗ ∈ − S∪S U and an algebraic map G0 : 0 (E U(n−)) , such that G0 = δ f on U → ⊗ ν ν ν 0. U∩U This statement then implies the existence of an algebraic extension of δν fν locally around every point of h∗ ( 1). This extension is unique and thus we obtain a − S∪S global algebraic extension on h∗ ( 1). Since codim( 1) 2wecanextend − S∪S S∪S ≥ this to a morphism G on the whole of h∗. Let now τ0 h∗ ( 1). ∈ − S∪S If τ0 / ,thenτ0 =h∗ ( ) and since f was algebraic on (Lemma ∈Hν ∈U − Hν∪S ν U 4), claim ( ) follows with 0 := and G0 := δ f . ∗ U U ν ν Let now τ0 ν. By assumption we then have τ0 ν ( 1), and Lemma 5 ∈H +∈H − S∪S implies that τ0,α∨ =t0 Zfor exactly one α R and µ0 = ν ν+ρ+τ0,α∨ α h i ∈ ∈ −h i is a weight of E with µ0 >ν. Again we construct a set h with a particular choice of hµ .Namely,choose U{ µ} { } for all weights µ P (E) with µ>νand µ = µ0 elements h h, such that ∈ 6 µ ∈ H (w (τ0 + ν),τ0)= w (τ0+ν) τ0 µ, h =0forallw .Forµ0however, µ · h · − − µi6 ∈W choose hµ0 such that 230 KAREN GUNZL¨

a) α, h =0and h µ0 i6 b) H (w (τ0 + ν),τ0)= w (τ0+ν) τ0 µ0,h =0foralls =w . µ0 · h · − − µ0i6 α 6 ∈W Lemma 5 ensures that such a choice of hµ is always possible. For τ h∗ we { } 1 ∈ set again pτ := µ P (E),µ>ν sym Hµ( ,τ), and ξ− (pτ )=:zτ Z.Forudefined ∈ − ∈ by u(τ )=pτ(τ+ν) we then obtain h = τ h∗ u(τ) =0 . According to Q U{ µ} { ∈ | 6 } Lemma 3 the map fν restricted to h is algebraic. Let us have a closer look U{ µ} at the map u : It vanishes along the hyperplane ker H , because for all τ h∗ α,t0 ∈ we have Hµ0 sα (τ + ν),τ = α, hµ0 τ,α∨ t0 = a Hα,t0 (τ), where a := α, h =· 0 according to assumption−h i· a).h i− · −h µ0i6 All the other hyperplanes along which u vanishes are not equal to ker Hα,t0 but do at most intersect it. Otherwise, (since τ0 ker H )wewouldhave ∈ α,t0 Hµ w (τ0+ν),τ0 = 0 for a pair (µ, w) =(µ0,sα) which is impossible by our choice· of the h . 6 µ Foru ¯ : h∗ k defined byuH ¯ = u, it follows by what we just said that → α,t0 u¯(τ0) = 0. With 0 := τ h∗ u¯(τ) =0 we then get 6 U { ∈ | 6 } τ0 0, • ∈U 1/u¯is algebraic on 0, • U h = 0 ker Hα,t0 ,thus 0= h ker Hα,t0 , •U{ µ} U − U U{ µ}∪ ( 0) h ,since ker Hα,t0 = . • U∩U ⊂U{ µ} U∩ ∅ If we define δ¯ = δ¯ : h∗ k by δH¯ = δ and set ν → α,t0 ν G0 : 0 E U ( n − ) U −→ ⊗ 1 τ δ¯(τ) u¯(τ) − z (e v ), 7→ τ ν ⊗ τ we know that G0 is algebraic on 0. In particular, G0 is algebraic on an open U neighborhood of τ0. This is now the local algebraic extension of δνfν around τ0 which we were looking for, because according to Lemma 3 we have for all τ h : ∈U{ µ} 1 δν(τ)fν(τ)=δν(τ)(u(τ))− zτ (eν vτ ) ⊗ 1 1 = δ¯ (τ)H (τ)(H (τ))− (¯u(τ))− z (e v ) ν α,t0 α,t0 τ ν ⊗ τ = G0(τ).

In particular, this equation holds for all τ ( 0) h , i.e. ( ). ∈ U∩U ⊂U{ µ} ∗ It remains to show that the set of zeros of G has codimension 2. ≥ In order to do this, let 2 be the union of all intersections of hyperplanes in ν with any other integral hyperplanes,S that is H

2 := 1 τ ν τ,α∨ = m Z for a pair (α, m) / ν . S S ∪{ ∈H |h i ∈ ∈N } We claim

( ) := τ h∗ G(τ)=0 ( 2) ∗∗ K { ∈ | }⊂ S∪S and conclude then that codim 2. Indeed, by deﬁnition the set 2 consists of a countable family of intersectionsK≥ of integral hyperplanes. On theS∪S other hand, G is algebraic on h∗ and therefore its set of zeros must be Zariski closed. Since a countable family of intersections of integral hyperplanes is Zariski closed only if it is a ﬁnite family of such intersections, the set must have codimension 2. K ≥ Let now τ0 h∗ ( 2). We have to show G(τ0) =0. ∈ − S∪S 6 If τ0 / ,wehaveδ (τ0)=0andτ0 =h∗ ( ). Now G equals δ f ∈Hν ν 6 ∈U − Hν ∪S ν ν on and hence G(τ0)=δ (τ0)f (τ0)=0. U ν ν 6 THE FINE STRUCTURE OF TRANSLATION FUNCTORS 231

Let now τ0 ν. Again by the ﬁrst part of the proof we know that there is ∈H 1 an open neighborhood 0 of τ0, such that G(τ)=δ¯(τ) u¯(τ) − z (e v ) for all U ν τ ν ⊗ τ τ 0.Sinceδ¯(τ0)= 0 it thus suﬃces to show ∈U ν 6 ( )’ z (e v ) =0. ∗∗ τ0 ν ⊗ τ0 6 In order to do this, we will use Lemmas 6 to 9 of the following section. Set e := eν and denote the projection onto the central character χ(τ0 + ν)bypr= pr : E U(n ) E U(n ). χ(τ0+ν) − − ⊗ ⊗ + Since τ0 ν ( 2), there exists exactly one α R such that τ0,α∨ Z. ∈H − S∪S ∈ h i∈ Therefore, Rτ0 := β R τ0,β∨ Z = α, α and hence the Weyl group { ∈ |h i∈ } { − } of this root system R equals = s β R = s .Forτ0and Wτ0 τ0 Wτ0 h β | ∈ τ0 i h αi µ0 = ν τ0 +ν+ρ, α∨ α we may then apply Lemma 8 and obtain a short exact sequence−h i

M(sα (τ0 + ν)) , pr E M(τ0) L(τ0 + ν) dim Eµ0 · → ⊗ M such that pr(e v )/ pr E M(τ0) generates the simple module L(τ0 + ν). The ⊗ τ0 ⊗ module pr E M(τ ) is projective (Lemma 7) and this forces the above short exact 0 sequence to be⊗ a nontrivial extension because otherwise, being a direct summand, L(τ0 + ν) would be projective too. This contradicts Lemma 7. Therefore (see for example [HS, Ch. 3, Lemma 4.1]) we have at least one direct summand M(sα (τ0 + ν)) of R2 := dim E M(sα (τ0 + ν)), such that the · µ0 · projection pro : R2 M(s (τ0 + ν)) induces a nontrivial extension α · L M(s (τ0 + ν)) , PO L(τ0 + ν) α · → together with a homomorphismpro ˜ : R1 := pr E M(τ0) PO, such that the ⊗ → following diagram commutes (the rows are exact, PO is a push-out of pro : R2 M(sα (τ0 + ν)) and R2 , R1; see for example the dual statement to [HS, Ch. 3,

Lemma· 1.3]): →

/ / R2 / R1 R1/R2 ∼= L(τ0 + ν)

pro pro˜ id

/ / M(s (τ0 + ν)) / PO PO /M(s (τ0 + ν)) = L(τ0 + ν) α · α · ∼ Since the bottom row is a nontrivial extension and since + = = s , Wτ0 ν Wτ0 h αi the push-out PO is isomorphic to P (τ0 + ν), the projective cover of L(τ0 + ν)in (Lemma 6). In the right loop of the above diagram lies the following element

diagram:O

pr(e vτ0 ) pr(e vτ0 )/R2 _

⊗_ ⊗

/ pro˜ pr(e v ) / pro˜ pr(e v ) /M(s (τ0 + ν)) ⊗ τ0 ⊗ τ0 α ·

Since pr(e v )/R2 is a generator of L(τ0 + ν) we conclude that ⊗ τ0 pro˜ pr(e v ) /M(s (τ0 + ν)) PO /M(s (τ0 + ν)) is also a generator of ⊗ τ0 α · ∈ α · L(τ0 +ν) and thus its preimagepro ˜ pr(e v ) generates the indecomposable mod- ⊗ τ0 ule PO = P (τ0 +ν). By Lemma 9 the element z is not contained in the annihilator ∼ τ0 of P (τ0 +ν), this means in particular 0 = z pro˜ pr(e v ) =˜pro z pr(e v ) 6 τ0 ⊗ τ0 τ0 ⊗ τ0 232 KAREN GUNZL¨

and we obtain zτ0 pr(e vτ0 ) = 0. Therefore, claim ( )’ is proven and the proof of Theorem 1 is ﬁnished⊗ provided6 we know that Lemma∗∗ 6, 7, 8 and 9 hold.

3.3. Proof of Lemmas 6, 7, 8 and 9. In the following let eν denote again the fixed extremal weight vector of the finite-dimensional irreducible g-module E = E(ν). Set v M(τ) the canonical generator of the Verma module and let E τ ∈ ⊗ U(n−) = E M(τ) for all τ h∗. The category is the category of all finitely ∼ ⊗ ∈ O generated g-modules, which are b-finite and semisimple over h [BGG2]. For λ h∗ ∈ set λ := w λ wλ P and Rλ := β R λ, β∨ Z . The group W { ∈W| − ∈ } { ∈ |h i∈ } λ is then the Weyl group to the root system Rλ [Ja1, 1.3]. Denote by P (λ)the projectiveW cover in of the simple module L(λ). O Lemma 6. Let λ h∗ with = s and M(λ)=L(λ)simple. If the short exact ∈ Wλ h αi sequence M(sα λ) , N L(λ) is a nontrivial extension, then N is isomorphic to P (λ). · →

Proof. For λ with λ = sα there exists in (up to isomorphism) a unique indecomposable projectiveW moduleh i P (λ), such thatO the short exact sequence M(s λ) , P (λ) L(λ) α · → is a nontrivial extension [BGG2, Ch. 4]. We thus have a nontrivial element of Ext1 L(λ),M(s λ) . The assertion follows immediately if we know that α · dim Ext1 L(λ),M(s λ) 1. Indeed, construct for M(s λ) , P (λ) L(λ) α· ≤ α · → the long exact homology sequence [HS, Ch. 3, Thm. 5.3] (set M = M(sα λ),P = P(λ),L=L(λ),Ext = Ext and let ω be the connecting homomorphism):· O Hom(M,M) ω Ext1(L, M) Ext1(P, M) Ext1(M,M) ···→ → → → →··· and note that Ext1(P, M)=0sinceP=P(λ) is projective [HS, Chap. 3, Prop. 2.6]. By exactness we obtain ω surjective. Since now for any Verma module MdimHom(M,M) = 1, we conclude that dim Ext1(L, M) 1. ≤ Lemma 7. Let τ h∗ with = s . ∈ Wτ h αi (a) If τ + ρ, α∨ 0,thenpr E M(τ) is projective, this means that h i≥ χ(τ+ν) ⊗ pr E M(τ) is a projective object of the category . χ(τ+ν) ⊗ O (b) If τ + ν + ρ, α∨ 0,thenM(τ+ν)=L(τ+ν)is simple and not projective. h i≤ + Proof. a) A Verma module M(λ) is projective if λ+ρ, β∨ 0 for all β Rλ R + h i≥ ∈ ∩ [Ja2, 4.8]. Since τ = sα ,wehaveRτ R = α and hence M(τ) is projective. Then also E MW(τ) ish projective,i because∩ for E{with} dim E< the functor ⊗ ∞

FE : MO−→OE M 7→ ⊗ maps projective objects of to projective objects of [BG]. Its direct summand pr E M(τ) E OM(τ) is then projective too.O χ(τ+ν) ⊗ ⊂ ⊗ b) A Verma module M(λ) is simple if and only if λ + ρ, β∨ 0 for all β + h i≤ ∈ Rλ R [Di, 7.6.24] or [Ja1, 1.8, 1.9]. Since τ+ν = τ = sα we obtain ∩ + W W h i R + R = α and the simplicity of M(τ + ν)=L(τ+ν) follows. Since τ ν ∩ { } the short exact sequence M(sα (τ + ν)) , P (τ + ν) L(τ + ν) is a nontrivial extension [BGG2, Ch. 4], the module· L(τ +→ν) cannot be projective. THE FINE STRUCTURE OF TRANSLATION FUNCTORS 233

Lemma 8. Let τ h∗ with = s and µ0 := ν τ+ν+ρ, α∨ α P (E) such ∈ Wτ h αi −h i ∈ that µ0 >ν. Then the module pr E M(τ) admits a chain of submodules χ(τ+ν) ⊗ pr E M(τ) = R1 R2 0, χ(τ+ν) ⊗ ⊃ ⊃ such that R2 = dim E M(sα (τ + ν)) and R1/R2 = M(τ + ν)=L(τ+ν)and ∼ µ0 · ∼ such that L(τ + ν) is generated by pr (e v ) /R2. L χ(τ+ν) ν ⊗ τ Proof. Let µ1,µ2,... ,µm be the weights of E without multiplicities, numbered such that µi <µj iνforces τ + ν + ρ, α∨ < 0and L χ(τ+ν) ν ⊗ τ h i together with + = = s this implies M(τ + ν)=L(τ+ν) (Lemma 7). Wτ ν Wτ h αi For µ h∗ deﬁne maps Hµ : h∗ h∗ k by Hµ(λ, τ):= λ τ µ, hµ λ, τ ∈ × → h − − 1i∀ ∈ h∗,hµ h.Forτ h∗set pτ := µ>ν,µ P (E) sym Hµ( ,τ)andzτ := ξ− (pτ ) Z ∈ ∈ ∈ − ∈ the preimage of p under the Harish-Chandra isomorphism ξ : Z ∼ ( h ∗ ) W·. τ Q −→ P

Lemma 9. Let τ0 h∗ with τ0 = sα and ρ, α∨ τ0,α∨ ν+ρ, α∨ 1. Assume the h h∈to be chosenW suchh thati −h i≤h i≤−h i− µ ∈ (a) α, h =0for µ0 := ν τ0 +ν+ρ, α∨ α and h µ0 i6 −h i (b) H w (τ0 + ν),τ0 =0for (w, µ) =(s ,µ0), w ,µ P(E) with µ>ν. µ · 6 6 α ∀ ∈W ∈ Then z / Ann P (τ0 + ν). τ0∈ Z Proof. We will ﬁrst give an explicit description of the annihilator AnnZ P (τ0 + ν) + by a theorem of Soergel [So]. For this let λ h∗, such that for all β R R we ∈ ∈ ∩ λ have λ + ρ, β∨ 0. Denote by w the longest element with respect to the h i≥ λ ∈Wλ Bruhat ordering. Then deﬁne for µ h∗ the map ∈ + µ : (h∗) ( h ∗ ) P p −→ Pµ+(p) 7→ + by µ (p) (τ ):=p(τ+µ) for all τ h∗. Then [So, 2.2]: ∈ 1 + + λ ξ− (p) Ann P (w λ) λ ( p ) ( h ∗ ) W ( h ∗ ) . ∈ Z λ · ⇐⇒ ∈ P P + Here (h∗) denotes the set of polynomial functions on h∗ without constant term. P We want to describe the annihilator of P (τ0 + ν) and choose for this λ0 := s α · (τ0 + ν)=τ0+µ0 =τ0+ν τ0 +ν+ρ, α∨ α.Wethenhave λ0 = s (τ+ν)= −h i W Wα· 0 τ0+ν = τ0 = sα and hence Rλ0 = α, α . By assumption we know that W W h i { +− } λ0 + ρ, α∨ 0 and obtain for all β R Rλ0 that λ0 + ρ, β∨ 0. Now wh = s isi≥ the longest element in ∈and we∩ may applyh Soergel’s theoremi≥ with λ0 α Wλ0 234 KAREN GUNZL¨

1 λ := λ0 = s (τ0 +ν). Assume we had ξ− (p) Ann P (w λ0)=Ann P(τ0+ν) α · ∈ Z λ0 · Z n λ + W 0 for p = i=1 piqi with pi (h∗) and qi (h∗). Then it follows for all pi that ∈P ∈P P + + p (λ0 + µ)= λ (p) (µ)= λ (p) (s µ)=p(λ0+s µ) µ h∗. i 0 i 0 i α i α ∀ ∈ For µ = α we obtainpi(λ0 + α)=pi(λ0 α) and this forces the derivative of p − in direction of α to vanish at the point λ0. Let us check this condition for pτ .By deﬁnition we have for all λ h∗: p (λ)= H (w λ, τ). If we deﬁnep ¯ ∈ τ µ>ν,w µ · τ byp ¯ H ( ,τ)=p ,weobtain ∈W τ µ0 − τ Q p¯(λ):= H (w λ, τ). τ µ · (w,µ)=(e,µ0) w Y6 ,µ>ν ∈W

Let pτ0 denote the derivative of pτ in direction α. By the product rule we have p0 =¯p0Hµ ( ,τ)+¯pτH0 ( ,τ). τ τ 0 − µ0 − Since s (τ0 + ν)=τ0+µ0 it follows that H s (τ0 + ν),τ0 =0andweget α · µ0 α · p0 sα (τ0+ν) =¯pτ sα (τ0+ν)H0 sα (τ0+ν),τ0 . But now neither of these τ0 0 µ0 two factors· is zero because· · p¯ s (τ0 + ν) = H w s (τ0 + ν),τ0 τ0 α · µ · α · (w,µ)=(e,µ0) w Y6 ,µ>ν ∈W = H w (τ0 + ν),τ0 µ · (w,µ)=(sα,µ0) w 6Y,µ>ν ∈W and by assumption (b) none of the factors H w (τ0 + ν),τ0 vanishes, hence µ · p¯τ sα (τ0 + ν) = 0. On the other hand, also H0 sα (τ0 + ν),τ0 =0sinceby 0 · 6 µ0 · 6 assumption (a) we have: H (λ, τ )= d λ + tα τ µ ,h = α, h =0. µ0 0 0 dt 0 0 µ0 µ0 |t=0 h − − i h i6 Together this implies p0 sα (τ0 + ν) = 0 and therefore ξ(pτ )=zτ cannot be τ0 · 6 0 0 contained in the annihilator of P (τ0 + ν). 4. The triangle function ∆ 4.1. Preliminaries. Let M be a representation of g and E a vector space. Then E M is a representation of g via X(e m):=e Xm for all X g,e E and m ⊗ M. If in addition E is also a representation⊗ ⊗ of g, then we obtain∈ a∈ second g-operation∈ on E M via X(e m):=Xe m+e Xm. To distinguish these two representations we⊗ denote the ﬁrst⊗ one by E⊗ˆ M.Letnow⊗ Ebe a ﬁnite-dimensional representation of g and ν P aweightofE⊗. ∈ Lemma 10. Let τ h∗ such that χ(τ +ν) = χ(τ +µ) for all µ P (E) with µ = ν. Then there exists a∈ unique natural isomorphism6 ∈ 6

can : E ˆ M(τ + ν) ∼ pr (E M(τ)), ν⊗ −→ χ(τ+ν) ⊗ such that can(e ˆv + )=pr (e v )for all e E . ⊗ τ ν χ(τ+ν) ⊗ τ ∈ ν Remark. For a generic weight, i.e. a weight τ h∗ such that τ,α∨ / Z for all α R, the central characters χ(τ + µ)forµ∈ P(E) are pairwiseh i distinct.∈ In particular,∈ in this case the condition of the lemma∈ is always satisﬁed. THE FINE STRUCTURE OF TRANSLATION FUNCTORS 235

Proof. By the so-called tensor identity we have a canonical isomorphism

U (E k ) ∼ E ( U k ) ⊗U(b) ⊗ τ −→ ⊗ ⊗ U ( b ) τ such that u (e a) u(e (1 a)). Call the left hand side F , the right ⊗ ⊗ 7→ ⊗ ⊗ hand side is E M(τ). Denote by µ1,... ,µ the weights of E. The filtration ⊗ m E M(τ)=M1 Mm 0, where the subquotients are isomorphic to direct sums⊗ of Verma modules⊃···⊃ (see Chapter⊃ 2), induces a filtration of F : U (E k )= ⊗U(b) ⊗ τ F=F1 F2 F 0 such that ⊃ ⊃···⊃ m ⊃ F /F +1 = U (E k ). j j ∼ ⊗U(b) µj ⊗ τ Now here the right hand side is canonically isomorphic to E ˆ M(τ + µ )(bymap- µj ⊗ j ping e uvτ+µj u (e 1)) and in particular, we have that χ (τ + µj)(Fj/Fj+1)= 0. ⊗ 7→ ⊗ ⊗ Let now ν = µ for a fixed i. By the condition on τ we know that χ(τ + ν) = i 6 χ(τ + µ ) for all j = i, j 1,... ,m and hence pr (F /F +1)=0for j 6 ∈{ } χ(τ+ν) j j all j = i.Thuswegetpr F=pr F1 =... =pr F and also 6 χ(τ+ν) χ(τ+ν) χ(τ+ν) i pr F +1 = ...=pr F = 0. We conclude that pr F F and that χ(τ+ν) i χ(τ+ν) m χ(τ+ν) ⊂ i F +1 ker(pr : F pr F ). Since pr (F /F +1)=F/F +1,we i ⊂ χ(τ +ν) i χ(τ+ν) χ(τ+ν) i i i i even know that Fi+1 is equal to this kernel. This now induces a natural isomorphism

F /F +1 ∼ pr F i i −→ χ(τ+ν) such that E ˆ M(τ + ν) ∼ pr F ν ⊗ −→ χ(τ+ν) e ˆ (uv + ) pr (u (e 1)). ⊗ τ ν 7−→ χ(τ+ν) ⊗ ⊗ We apply the tensor identity F = E M(τ) and the lemma follows. ∼ ⊗ Let us recall the theorem of Bernstein and Gelfand [BG] for projective functors. For this denote by the category of all Z-ﬁnite g-modules and for χ MaxZ let M ∈ (χ):= M χM =0 , M { ∈M| } n ∞(χ):= M for all m M exists n N such that χ m =0 . M { ∈M| ∈ ∈ } A projective χ-functor is then a functor F : (χ) , which is isomorphic to a direct summand of a functor E for a ﬁnite-dimensionalM →M representation ⊗ τ+ν E. In particular, the restriction of the translation functor T : ∞ χ(τ) τ M → ∞ χ(τ + ν) to the subcategory (χ(τ)) is a projective χ(τ)-functor. For two M ˜ M ˜ projective χ-functors F, F : (χ) denote by Hom (χ) (F, F )thespaceof M →M M → all natural transformations from F to F˜. Theorem ([BG, 3.5]). Let F, F˜ : (χ) be projective χ-functors and let M →M τ h∗ such that χ(τ)=χand M(τ) is projective. Then the obvious map ∈ ˜ ˜ Hom (χ(τ )) (F, F ) ∼ Homg(FM(τ),FM(τ)) M → −→ is an isomorphism.

Remark. (i) The Verma module M(τ) is projective if and only if τ + ρ, α∨ / 1, 2,... for all α R+. In particular, for a generic weighthτ the Vermai ∈ {−module− M(τ}) is always∈ projective. (ii) Note that in [BG] the Verma module with highest weight τ is denoted by Mτ+ρ. Accordingly, the theorem there is formulated for all τ with τ,α∨ / 1, 2,... for all α R+. h i ∈ {− − } ∈ 236 KAREN GUNZL¨

4.2. Deﬁnition of ∆. For ν P let E(ν) be the ﬁnite-dimensional irreducible ∈ g-module with extremal weight ν and let x . For a weight τ h∗ with χ(τ + ν) = χ(τ + µ) for all µ P (E) with µ∈W= ν, Lemma 10 yields a∈ canonical isomorphism6 ∈ 6

+ E(ν) ˆ M(x (τ + ν)) ∼ pr (E(ν) M(x τ)) = T τ νM(x τ). xν ⊗ · −→ χ(τ+ν) ⊗ · τ · Let now ν, µ P be integral weights. We set E0 := E(ν), E00 := E(µ)and E:= E(ν + µ).∈ For generic τ consider the following sequence of isomorphisms:

τ+ν+µ τ+ν τ+ν+µ Hom (χ(τ )) Tτ+ν Tτ ,Tτ M → ◦ τ +ν+µ + + + ∼ Homg T T τ νM(x τ),Tτ ν µM(x τ) −→ τ +ν τ · τ · ∼ Homg E00 ˆ E0 ˆ M(x (τ + ν + µ)),Ex(µ+ν)ˆM(x (τ +ν+µ)) −→ xµ⊗ xν ⊗ · ⊗ · ∼ Homk(E00 E0 ,Ex(µ+ν)) −→ xµ ⊗ xν ∼ E 00∗ E0∗ E . −→ xµ ⊗ xν ⊗ x(µ+ν) Here, we obtain the ﬁrst isomorphism by the Theorem of Bernstein-Gelfand, the second is due to Lemma 10, the others are obvious. We call this map nat(µ, ν; x)(τ)

and deﬁne for generic τ and for the triangle

ν = µ

ν+µ / τ / the value of the triangle function ∆ by

1 1 ∆(µ, ν; x)(τ):=det x− nat(µ, ν; x)(τ) (nat(µ, ν; e)(τ))− . ◦ ◦ We have yet to explain the map

1 x− : E(µ)∗ E(ν)∗ E(µ + ν) E(µ)∗ E(ν)∗ E(µ + ν) + . xµ ⊗ xν ⊗ x(ν+µ) → µ ⊗ ν ⊗ ν µ For this let G be a simply connected algebraic group with Lie algebra g and T G a maximal torus with Lie algebra h. Each ﬁnite-dimensional representation ⊂ E of g is in a natural way a representation of G. The operation of NG(T ), the normalizer of T in G,onEstabilizes E0 and factors over an operation of = 1 W N (T )/T .Themapx− is given by this operation of on the zero weight space G W (E(µ)∗ E(ν)∗ E(ν + µ))0. Now⊗ take for⊗τ not only any, but rather the generic weight: For this denote by S := Sk(h) the symmetric algebra of h and let K := Quot(S)beitsquotient ﬁeld. We then have a k-linear map h , S, Kand obtain thus a K-linear map τ : K h K such that the following→ diagram→ commutes ⊗k −→ h S −−−−→

 τ  K  k h K ⊗y −−−−→ y With this τ (= tautologous) we then obtain ∆(µ, ν; x)(τ) K×. These are the triangle functions. ∈ THE FINE STRUCTURE OF TRANSLATION FUNCTORS 237

4.3. Uncanonical definition of ∆. We defined the triangle functions by a series of canonical isomorphisms. For our purposes it is sometimes more convenient to realize the triangle functions in the following—uncanonical—way: Let ν P , x ∈ ∈ and choose a fixed extremal weight vector 0 = eν E(ν)ν . Since extremal Wweight spaces are one dimensional, this choice is unique6 ∈ up to non-zero scalar. By Lemma 10 we obtain for generic τ h∗ a—no longer canonical—isomorphism ∈ F (x τ): M(x (τ+ν)) ∼ T τ + ν M ( x τ ) xν · · −→ τ · v x ( τ + ν ) prχ(τ+ν)(˙xeν vx τ ). · 7→ ⊗ · Herex ˙ G denotes a pre-image of x =N T/T.Wehave˙xe E(ν) . ∈ ∈W∼ G ν ∈ xν Let now µ P be another weight. Choosee ˜µ E(µ)µ ande ¯µ+ν E(µ + ν)µ+ν , both non-zero,∈ and consider for generic τ the following∈ sequence of∈ isomorphisms:

τ+ν+µ τ+ν τ+ν+µ Hom (χ(τ )) Tτ+ν Tτ ,Tτ M → ◦ τ +ν+µ + + + ∼ Homg T T τ νM(x τ),Tτ ν µM(x τ) −→ τ +ν τ · τ · ∼ Homg M(x (τ + ν + µ)),M(x (τ +ν+µ)) −→ · · ∼ k. −→ Denote this map by Nat(µ, ν; x)(τ). Here again the ﬁrst isomorphism is clear by the Theorem of Bernstein-Gelfand, the second is due to the maps F (x τ) x(ν+µ) · and F (x (τ + µ)) T τ +ν+µF (x τ). We then obtain xµ · ◦ τ +ν xν · 1 ∆(µ, ν; x)(τ)=Nat(µ, ν; x)(τ) (Nat(µ, ν; e)(τ))− (1). ◦ It follows that this characterization of ∆ is independent of the choice of the weight vectors eν , e˜µ ande ¯µ+ν, and also independent of the choice of the pre-imagex ˙ of x.

5. Bernstein’s relative trace and a special case of ∆

5.1. The relative trace. Let us recall the definition of the relative trace trE defined by Bernstein in [Be]. Let g-mod be the category of all g-modules and let E be a finite-dimensional g-module. Denote by FE : g-mod g-mod the functor defined by F (M ):=E M. The relative trace tr :End → (F ) End (Id) is E ⊗ E g-mod E → g-mod then a morphism from the endomorphisms of the functor FE to the endomorphisms of the identity functor on g-mod, defined by M trE :Endg(E M) Endg M a ⊗ → trM (a) 7→ E where

M i id a cont id tr (a): M E ∗ E M ⊗ E ∗ E M ⊗ M . E −−−−→ ⊗ ⊗ −−−−→ ⊗ ⊗ −−−−−→ j c Here,iisthemapM, Endg(E) M E∗ E M with j(m):=idE m → ⊗ → ⊗ ⊗ ⊗ and c the canonical isomorphism Endg(E) ∼= E∗ E.Themapcont:E∗ E k denotes the evaluation map. ⊗ ⊗ → Bernstein has calculated an explicit formula for the relative trace, by considering it as a map from (h∗)W· to itself in the following way: First, we identify P Endg-mod(Id) ∼= Z with the center of the enveloping algebra U,thenwemake use of the natural morphism Z End (F ) and composing this with the → g-mod E trace map we obtain tr : Z Z. By means of the Harish-Chandra isomorphism E → 238 KAREN GUNZL¨

ξ : Z ∼ ( h ∗ ) W· (normalized by z ξ(z) Un) we may then regard tr as an −→ P − ∈ E endomorphism of (h∗)W·. P Deﬁne now on (h∗) a convolution f P (E) f by P 7→ ∗ (P (E) f )(λ):= f(λ+µ), ∗ µ P(E) ∈X where the sum is taken over all weights µ P (E) with their multiplicities. Set ∈ Λ(λ):= α R+ λ+ρ, α∨ .Thenwehave ∈ h i 1 TheoremQ([Be]). tr (f)=Λ− (P(E) Λf)for all f (h∗)W·. E ∗ ∈P If we choose now for M the Verma module M(λ) we can associate to each endomorphism f End (E M(λ)) an element trM(λ)(f)ofEnd (M(λ)) k.As g E g ∼= an endomorphism∈ of M(λ)⊗ this element operates on M(λ) by multiplication with M(λ) the scalar (trE (f))(λ) and we obtain by Bernstein’s Theorem for all λ h∗ and for all w ∈ ∈W M(w λ) 1 tr · (f) (λ)= Λ− (P(E) Λf) (λ) E ∗ 1 =(Λ( λ))− Λ(λ + µ)f(λ + µ). µ P (E) ∈X

5.2. The special case ∆( ν, ν; w0). Let now E = E(ν), w0 the longest − ∈W element and pr Endg(E(ν) M(w0 τ)) the projection on the central χ(τ+ν) ∈ ⊗ · M(w0 τ) character χ(τ +ν). Then tr · (pr ) is an element in Endg(M(w0 τ)) = k E(ν) χ(τ +ν) · ∼ and we have

+ Theorem 2. Let ν P be a dominant weight and τ h∗ generic. Then ∈ ∈ M(w0 τ) ∆( ν, ν; w0)(τ)=tr · (pr ). − E(ν) χ(τ +ν) Proof. Postponed to 5.3.

Regarding ∆( ν, ν; w0) as a polynomial function on h∗ we obtain for this special case an explicit formula:−

+ Corollary. Let ν P be a dominant weight and τ h∗ generic. Then ∈ ∈ τ+ν+ρ, α∨ ∆( ν, ν; w0)(τ)= h i . − τ + ρ, α∨ α R+ Y∈ h i Proof. Under the morphism (h∗)W· = Z End (F ) the projection P ∼ → g-mod E prχ(τ+ν) is the image of a polynomial function, which takes value 1 at all weights λ (τ+ν) and vanishes at all other weights τ + µ with µ P (E). Call this∈W· polynomial functionpr. ¯ By Bernstein’s formula for the relative∈ trace we then obtain

M(w0 τ ) 1 · trE(ν) (¯pr) (τ)=(Λ(τ))− Λ(τ + µ)¯pr(τ + µ) µ P (E(ν)) ∈ X and by deﬁnition ofpr ¯ the value ofpr( ¯ τ + µ)forµ P(E) does not vanish if and only if there is a w such that w (τ + µ)=(τ+∈ν). Since τ is generic, this is only possible for w ∈W= e and hence µ =· ν. Inthiscasewehave¯pr(τ + ν)=1and THE FINE STRUCTURE OF TRANSLATION FUNCTORS 239 since furthermore dim E(ν)ν =1,weget µ P(E(ν)) Λ(τ + µ)¯pr(τ + µ)=Λ(τ+ν). The claim now follows by Theorem 2 and the∈ equation P M(w τ ) Λ(τ + ν) τ + ν + ρ, α∨ 0· trE(ν) (¯pr) (τ)= = h i . Λ(τ) τ + ρ, α∨ α R+ h i Y∈

5.3. Proof of Theorem 2. First we make some more general preliminary remarks.

τ +ν τ τ τ+ν 5.3.1. The adjunctions (Tτ ,Tτ+ν) and (Tτ +ν ,Tτ ). Let and be categories and F : ,G: two functors. Then an adjunctionA (F,B G)ofF and G is a familyA→B of isomorphismsB→A

(F, G)M,N := (F, G):Hom (FM,N) ∼ Hom (M,GN), B −→ A which is natural in M and N (M , N ). For example, for E a ﬁnite-dimensional∈A ∈Bg-module we obtain an adjunction (F ,F )ofF :g-mod g-mod and F : g-mod g-mod as the composition E E∗ E → E∗ → Homg(E M,N) Homg(E∗ E M,E∗ N) Homg(M,E∗ N). ⊗ → ⊗ ⊗ ⊗ → ⊗ Here, the ﬁrst map is given by f id f, the second by g g i. Interchanging 7→ E∗ ⊗ 7→ ◦ E and E∗ we obtain in the same way an adjunction (FE∗ ,FE). Its inverse is the composition

Homg(M,E N) Homg(E∗ M,E∗ E N) Homg(E∗ M,N), ⊗ → ⊗ ⊗ ⊗ → ⊗ where the first map is again given by f idE∗ f and the second by g (cont id ) g.Letnowν Pbe an7→ integral⊗ weight. Each identification7→ ⊗ N ◦ ∈ ϕ : E(ν)∗ ∼ E ( ν ) then defines adjunctions (FE(ν),FE( ν))and(FE( ν),FE(ν)). More precisely,−→ we− have − −

(FE(ν),FE( ν)): Homg(E(ν) M,N) ∼ Homg(M,E( ν) N) − f ⊗ −→ (ϕ f)− i ⊗ 7→ ⊗ ◦ and 1 (FE( ν),FE(ν))− :Homg(M,E(ν) N) ∼ Homg(E( ν) M,N) − ⊗ −→ − ⊗ 1 g (cont id ) (ϕ− g). 7→ ⊗ N ◦ ⊗ Let now i : ∞(χ) , denote the embedding functor. We then have in a χ M →M natural way adjunctions (iχ, prχ)and(prχ,iχ). Since for the translation functor τ+ν τ+ν τ Tτ =prχ(τ+ν) FE(ν) iχ(τ), we thus obtain also adjunctions (Tτ ,Tτ+ν)and τ τ+ν ◦ ◦ (Tτ+ν,Tτ ).

1 2 5.3.2. The natural transformations adj and adj . Let M ∞(χ(τ)) and con- τ +ν τ+ν ∈M τ+ν τ sider Id Homg(Tτ M,Tτ M). By means of the two adjunctions (Tτ ,Tτ+ν) τ ∈ τ+ν 1 and (Tτ+ν,Tτ )− we get two maps as the images of Id: 1 2 adj Homg(M,T τ T τ+νM)andadj Homg(T τ T τ +νM,M) M ∈ τ+ν τ M ∈ τ +ν τ and one checks that we obtain in this way natural transformations adj1 Hom (Id,Tτ Tτ+ν)andadj2 Hom (T τ T τ+ν, Id). By∈ ∞(χ(τ )) τ+ν τ ∞(χ(τ )) τ+ν τ composingM these→ two maps,◦ we get a canonical∈ endomorphismM → of M◦:

adj1 adj2 M M T τ T τ + ν M M M. −−−−→ τ + ν τ −−−−→ 240 KAREN GUNZL¨

We want to describe this endomorphism in more detail. By deﬁnition, the adjunction (T τ+ν,Tτ ) is just the composition of adjunctions (i , pr ) (F , τ τ+ν χ(τ) χ(τ) ◦ E(ν) FE( ν)) (prχ(τ +ν), iχ(τ +ν)) and one checks easily that the image of − ◦ Id Homg(T τ +νM,T τ+νM) ∈ τ τ under (prχ(τ+ν), iχ(τ+ν)) is precisely the projection prχ(τ+ν). Thus, the image of Id τ+ν τ under the adjunction (Tτ ,Tτ+ν) can be described by the composition

1 i ϕ pr τ τ + ν adjM : M E ∗ E M ⊗ T T M. −−−−→ ⊗ ⊗ −−−−→ τ + ν τ 2 Here, we put E = E(ν) and pr = prχ(τ+ν). Analogously, we obtain for adjM the composition

2 1 adj : τ τ+ν ϕ− pr cont idM M T T M ⊗ E ∗ E M ⊗ M τ+ν τ −−−−−→ ⊗ ⊗ −−−−−−−→ and taken together we have the following commutative diagram:

1 2

adjM adjM

/ / / τ τ+ν

M T + T M M O

τ ν τ O

i ϕ idE M cont idM

⊗ ⊗ ⊗

idE pr idE pr

/ / ∗ ⊗ / ∗ ⊗ E∗ E M E∗ E M E∗ E M ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ We thus obtain for adj2 adj1 M ◦ M id pr i E∗ ⊗ χ(τ+ν) cont idM M E ∗ E M E ∗ E M ⊗ M. −−−−→ ⊗ ⊗ −−−−−−−−−−→ ⊗ ⊗ −−−−−−−→ Comparing this with the relative trace trE for E = E(ν), it follows immediately for all M ∞(χ(τ)) that ∈M 2 1 adj adj =trM (pr ) Endg(M) M ◦ M E(ν) χ(τ +ν) ∈ or, regarded as a natural transformation of the identity functor Id : ∞(χ(τ)) M ∞(χ(τ)) to itself →M adj2 adj1 =tr (pr ). ◦ E(ν) χ(τ +ν)

Let now ιχ : (χ) , denote the embedding functor, let F, G : ∞(χ) be functors andM denote→M by F (χ), resp. G(χ) its restrictions to theM subcategory→M (χ). Each natural transformation n from F to G can be regarded as natural M transformation from F (χ)toG(χ) by ﬁrst applying ιχ and then n. In particular, 1 we obtain in this way the two natural transformations adj Hom (χ(τ )) Id(χ), τ τ+ν 2 τ ∈ τ+νM → Tτ+ν Tτ (χ) and similarly adj Hom (χ(τ )) Tτ+ν Tτ (χ), Id(χ) . ◦ ∈ M → ◦ 5.3.3. We have ∆( ν, ν; w )(τ)=adj2 adj1 . By choosing for M the 0 M(w0 τ) M(w0 τ) − · ◦ · 2 Verma module M(τ), we can assign to each τ h∗ a canonical element adj ∈ M(τ) ◦ adj1 End(M(τ)) = k. We will see in the following, that for dominant ν this M(τ) ∈ ∼ is precisely the triangle function ∆( ν, ν; w0)(τ). For x and τ generic we call τ τ+ν− ∈W 1 φx(x τ) the isomorphism Tτ+ν Tτ M(x τ) ∼ M ( x τ ) given by (φx(x τ))− = τ · · −→ · · Tτ +νFxν(x τ) F xν(x (τ + ν)) (see 4.3). First we show that for dominant ν and · ◦ − · THE FINE STRUCTURE OF TRANSLATION FUNCTORS 241 generic τ the following diagram commutes:

adjM(τ) /

T τ T τ+νM(τ) / M(τ) 5

τ+ν τ 5

k k

φe(τ) k

k k

k id

k k M(τ)

Let again denote e E(ν) the fixed extremal weight vector for a dominant ν ∈ ν integral weight ν.Sincew0is the longest element in the Weyl group, the dominance of ν implies that e ν :=w ˙ 0eν is a weight vector of weight ν. Here,w ˙ 0 G is a − − ∈ representative of w0 N (T )/T . ∈ G Define now a pairing E( ν) E(ν) k by e ν ,eν := 1 and obtain thus an − × → h − i identification ϕ : E(ν)∗ ∼ E ( ν ). Let v M(τ) be the canonical generator. To −→ − τ ∈ see that the above diagram commutes, it suffices to show that the pre-image of vτ τ τ+ν 2 in Tτ+ν Tτ M(τ) is mapped again to vτ when applying adjM(τ).Wehave

1 (φe(τ))− (vτ )=prχ(τ) e ν (prχ(τ +ν)(eν vτ )) − ⊗ ⊗ =prχ(τ)(e ν eν vτ) − ⊗ ⊗ =e ν eν vτ prχ(e ν eν vτ ). − χ=χ(τ) − ⊗ ⊗ − 6 ⊗ ⊗ M The ﬁrst equation holds by deﬁnition of φe(τ), the second follows since for dominant ν we have prχ(τ+ν)(eν vτ )=eν vτ (see the example in 3.1) and the third ⊗ ⊗ 2 equation is just a direct sum decomposition. Since adjM was the composition

1 τ τ+ν ϕ− pr cont idM T T M ⊗ E ∗ E M ⊗ M. τ+ν τ −−−−−→ ⊗ ⊗ −−−−−−−→ 2 we know in particular that adjM(τ) is a g-module homomorphism to M(τ). Then it 2 is clear, that adjM(τ) maps χ=χ(τ ) prχ(e ν eν vτ ) to zero since this element has 6 − ⊗ ⊗ the wrong central character. For e ν eν vτ we use the fact that e ν ,eν =1and L 2 − ⊗ ⊗ h − i 2 obtain as an image under adjM(τ) precisely vτ . Taken together, we get adjM(τ) = id φe(τ), i.e. the above diagram commutes. ◦This now means that under the map Nat( ν, ν; e)(τ) the image of adj2 τ τ+ν τ − ∈ Hom (χ(τ )) Tτ+ν Tτ (χ),Tτ (χ) is just the identity id End(M(τ)). By meansM of the→ uncanonical◦ deﬁnition of the triangle function we∈ deduce 1 ∆( ν, ν; w0)(τ) = Nat( ν, ν; w0)(τ) Nat( ν, ν; e)(τ) − (id ) − − ◦ − M(τ ) 2 = Nat( ν, ν; w0)(τ) (adj ) − and the Theorem of Bernstein-Gelfand implies that the right hand side of the following diagram commutes:

adj1 adj2

M(w0 τ) M(w0 τ)

/ / · / τ τ+ν ·

( ) M(w0 τ) Tτ+νTτ M(w0 τ) M(w0 τ)

U 4

U i U

∗ · · i ·

U i

i U

( ) i U

φw w0 τ i U

0 i

U i U

· i id i

U ∆( ν,ν;w0)(τ)

U i

* i − M(w0 τ) · 242 KAREN GUNZL¨

The commutativity of the left hand side can be shown in an analogous way. We thus obtain for dominant ν

2 1 adjM(w τ) adjM(w τ) =∆( ν, ν; w0)(τ) 0· ◦ 0· − 2 1 M(w τ) and together with the equation adj adj =tr 0· (pr ) from M(w0 τ) M(w0 τ) E(ν) χ(τ +ν) · ◦ · section 5.3.2 we deduce Theorem 2.

5.4. The case M(τ) T τ T τ + ν M ( τ ) M ( τ ) . −→ τ + ν τ −→ + Corollary of the proof. Let ν P be dominant and τ h∗ generic. Then the composition ∈ ∈

adj1 adj2 M(τ) M(τ) T τ T τ + ν M ( τ ) M(τ) M ( τ ) −−−−−→ τ + ν τ −−−−−→

τ+ν+ρ,α∨ is just multiplication with s(τ):= + h i ,wheresis considered as an α R τ +ρ,α∨ element in Quot(S(h)). ∈ h i Q Proof. The commutativity of the diagram ( ) in the proof of Theorem 2 is equivalent to the commutativity of ∗

1 2

adjM(τ) adjM(τ)

/ /

M(τ) / T τ T τ+νM(τ) M(τ)

S 5

τ+ν τ 5

S k

k S

φe(τ) k

S k

S k S

k id S

∆( ν,ν;w0)(τ) k

S k

) − k M(τ)

2 1 τ+ν+ρ,α∨ Hence (adj adj )M(τ) =∆( ν, ν; w0)(τ)= α R+ h τ +ρ,α i = s(τ). ◦ − ∈ h ∨i Q 6. Calculation of ∆

Theorem 3. For λ h∗ set α¯(λ):=1if λ, α∨ < 0 and α¯(λ):=0if λ, α∨ 0. ∈ h i h i≥ Let ν, µ P be integral weights and x . Then there exists a constant c k×, independent∈ of τ,ν,µ and x, such that ∈W ∈ τ,α α¯(ν) τ + ν, α α¯(µ) τ + ν + µ, α α¯(ν+µ) ∆(µ, ν; x)(τ ρ)=c ∨ ∨ ∨ . h iα¯(ν+hµ) i α¯(hν) i α¯(µ) − τ,α∨ τ + ν, α∨ τ + ν + µ, α∨ α R+ ∈ h i h i h i with Yxα R ∈ − Proof. Postponed.

For an integral weight ν P deﬁne the map δ (h∗)asinSection3.2by ∈ ν ∈P

δν(τ):= ( τ + ρ, α∨ nα). + h i− α R 0 nα< ν,α Y∈ ≤ Y−h ∨i Lemma 11. Let π(τ ) be the product on the right hand side of the equation in Theorem 3. Then δ (τ)δ (τ+ν)δ (x τ) π(τ + ρ)= ν µ x(ν+µ) · . ±δ (x τ)δ (x (τ + ν))δ + (τ) xν · xµ · ν µ THE FINE STRUCTURE OF TRANSLATION FUNCTORS 243

Proof. Let x be ﬁxed and let α R+ be a positive root. We then have in ∈W ∈ δν (τ)thefactor ( τ + ρ, α∨ n). Suppose now that xα is also a 0 n< ν,α∨ h i− positive root. Then≤ we obtain−h i in δ (x τ)thefactor Q xν · ( x τ + ρ, xα∨ n)= ( x(τ + ρ),xα∨ n) h · i− h i− 0 n< xν,xα 0 n< ν,α ≤ −hY ∨i ≤ Y−h ∨i = ( τ + ρ, α∨ n) h i− 0 n< ν,α ≤ Y−h ∨i + and hence in the quotient δν (τ)/δxν (x τ) all the products over α R with xα R+ cancel out. Let now α R+ such· that xα / R+, i.e. xα is∈ a positive root.∈ In this case we have in δ (x∈ τ)thefactor ∈ − xν · ( x τ + ρ, xα∨ n)= ( τ + ρ, α∨ n) h · − i− −h i− 0 n< xν, xα 0 n< ν,α ≤ −hY− ∨i ≤ Yh ∨i ν,α∨ =( 1)h i ( τ + ρ, α∨ + n) − h i 0 n< ν,α ≤ Yh ∨i and taken together we get

( τ + ρ, α∨ nα) δν(τ) ν,α∨ 0 nα< ν,α∨ = ( 1)h i ≤ −h i h i− . δxν(x τ) − ( τ + ρ, α∨ + nα) α R+ Q 0 nα< ν,α∨ · ∈ ≤ h i h i withYxα R ∈ − Q Considering then the diﬀerent cases according to whether ν, α∨ , µ, α∨ and h i h i ν + µ, α∨ are positive or negative, we obtain the closed formula h i δ (τ)δ (τ + ν)δ (x τ) ν µ x(ν+µ) · = επ(τ+ρ), δ (x τ)δ (x (τ + ν))δ + (τ) xν · xµ · ν µ (¯α(ν) ν,α∨ +¯α(µ) µ,α∨ α¯(ν+µ) µ+ν,α∨ ) where ε := ( 1) h i h i− h i = 1. α R+ − ± ∈ with xα R Q∈ − Now, let τ h∗ be a generic weight and recall (see 4.3) the map Natx(τ):= Nat(µ, ν; x)(τ):∈

τ+ν+µ τ+ν τ+ν+µ Hom (χ(τ )) Tτ+ν Tτ ,Tτ M → ◦ τ +ν+µ + + + ∼ Homg T T τ νM(x τ),Tτ ν µM(x τ) −→ τ +ν τ · τ · ∼ Homg M(x (τ + ν + µ)),M(x (τ +ν+µ)) −→ · · ∼ k. −→ Denote the pre-image of 1 k under this isomorphism by the natural transfor- x 1 ∈ τ+ν+µ τ+ν τ+ν+µ mation G (τ) := (Natx(τ))− (1) Hom (χ(τ )) (Tτ+ν Tτ ,Tτ ). ∈ M → ◦ e x Lemma 12. For generic τ h∗ we have G (τ)=∆(µ, ν; x)(τ)G (τ). ∈ 1 Proof. By deﬁnition, ∆(µ, ν; x)(τ)=Nat (τ) (Nat (τ))− (1) and therefore x ◦ e e 1 1 G (τ) = (Nat (τ))− Nat (τ) (Nat (τ))− (1) x ◦ x ◦ e 1 = (Natx(τ))− (∆(µ, ν; x)(τ)) 1 =∆(µ, ν; x)(τ)(Natx(τ))− (1) =∆(µ, ν; x)Gx(τ). 244 KAREN GUNZL¨

n Let now integral weights ν1,... ,νn begivensuchthat i=1 νi =0.Weset the translation functor T (ν ):=Tτ+ν1+ +νi when there is no ambiguity of i τ+ν1+ ···+νi 1 ··· − P the respective categories. For λ P an integral weight and τ generic there are ∈ isomorphisms (see 4.3) F (x τ):M(x (τ+λ)) ∼ T τ + λ M ( x τ ), such that xλ · · −→ τ · (Fxλ(x τ))(vx τ )=prχ(τ+λ)(˙xeλ vx τ ). Here, eλ E(λ)λ is a fixed chosen ex- · · n ⊗ · ∈ tremal weight vector. Since i=1 νi = 0, we can compose these isomorphisms to obtain an isomorphism M(x τ) = T (νn) T(ν1)M(x τ) and thus also ·P ∼ ··· · Homg M(x τ),M(x τ) ∼ Homg M(x τ),T(νn) T(ν1)M(x τ) · · −→ · ··· · , Hom U(n−),E(ν ) E(ν1) U(n−) , → k n ⊗···⊗ ⊗ where we have identified the Verma module M(x τ) with U(n−). Call now the x· image of the identity on M(x τ) under this map h (ν1,... ,νn)(τ) and let h∗ be the set of all generic weights.· We then obtain for all x a function U⊂ ∈W x h (ν1,... ,νn): Homk U(n−),E(νn) E(ν1) U(n−) U→ x ⊗···⊗ ⊗ τ h (ν1,... ,ν )(τ), 7→ n x such that h (ν1,... ,νn)(τ) maps the element vx τ M(x τ) = U(n−)tothe · ∈ · ∼ element prχ(τ+ν + +ν )(˙xeνn ( prχ(τ+ν )(˙xeν1 vx τ )) ) for fixed vectors 1 ··· n ⊗ ···⊗ 1 ⊗ · ··· e E(ν ) . νi ∈ i νi x Lemma 13. Set d (ν1,... ,νn)(τ):=δxν1 (x τ)δxν2 (x (τ + ν1)) δxνn (x (τ + x · x · ··· · ν1 + +νn 1)). Then the map d (ν1,... ,νn)h (ν1,... ,νn) is algebraic on and ··· − U there exists an algebraic extension on h∗ whose set of zeros has codimension 2. ≥ Remark. Here, we call a map a : Wto a vector space W algebraic,ifitisa morphism of varieties. Of course,U→ this is defined only if dim W< .Inourcase x ∞ however, the image of h (ν1,... ,νn) is always contained in a finite-dimensional subspace of Homk U(n−),E(ν) E(ν1) U(n−) and we may thus regard x ⊗···⊗ ⊗ h (ν1,... ,ν ) as a map between varieties. n Proof. Let ν P be an integral weight and recall the maps ∈ fν : h∗ E ( ν ) M ( τ ) ∼= E ( ν ) U ( n − ) τ −→ ⊗pr (e v ⊗). 7→ χ(τ+ν) ν ⊗ τ Let now x be fixed. For generic τ define then the map ax(τ)by ∈W ν x a (τ): M(x τ)=U(n−) E ( ν ) M ( x τ ) = E ( ν ) U ( n − ) ν · ∼ −→ ⊗ · ∼ ⊗ v x τ δxν(x τ)fxν (x τ) · 7→ · · where fxν(x τ)=prχ(τ+ν)(˙xeν vx τ ). Since fν is algebraic on (see Theorem · ⊗ · U 1), we obtain in this way also an algebraic map x aν : Homk U(n−),E(ν) U(n−) U−→τ ax(τ). ⊗ 7→ ν x Here again, the image of aν is contained in a finite-dimensional subspace of x Homk U(n−),E(ν) U(n−) and we regard aν in this way as a map between varieties. ⊗ According to Theorem 1 there is an algebraic extension of δνfν on h∗,which x vanishes only on a set of codimension 2. Therefore, also aν has such an algebraic x ≥ x extension and we call it again aν . For generic τ we then have (aν (τ))(vx τ )= · THE FINE STRUCTURE OF TRANSLATION FUNCTORS 245

x δxν(x τ)prχ(τ+ν)(˙xeν vx τ ). Note that for all τ h∗ the vector (aν (τ))(vx τ ) · ⊗ · ∈ · ∈ E(ν) M(x τ) generates a Verma module with highest weight x (τ + ν). The ⊗ x · · image of aν (τ) is thus always contained in a Verma module M(x (τ + ν)) E(ν) x · ⊂ ⊗ M(x τ) and we can identify the element (aν (τ))(vx τ ) with the canonical generator · · vx (τ+ν) M(x (τ + ν)). Using the isomorphism M(x (τ + ν)) ∼= U(n−), we may then· apply∈ the· map ax (τ + ν) for another weight ν · P . In particular, we can ν0 0 concatenate the maps ax (τ + ν + +ν ) ax∈(τ+ν ) ax (τ)andobtain νn 1 n 1 ν2 1 ν1 in this way an algebraic map ··· − ◦···◦ ◦

a : h∗ Homk U(n−),E(νn) E(ν1) U(n−) −→ x ⊗···⊗ x ⊗ x τ aν (τ + ν1 + +νn 1) aν (τ +ν1) aν (τ) 7→ n ··· − ◦···◦ 2 ◦ 1 which vanishes only on a set of codimension 2 and which maps a generic weight τ ≥ x to a(τ)=δxν1 (x τ)δxν2 (x (τ +ν1)) δxνn (x (τ +ν1+ νn 1)) h (ν1,... ,νn)(τ)= x · x · ··· · ··· − d (ν1,... ,νn)(τ)h (ν1,... ,νn)(τ). We thus obtain the map a as the desired algebraic extension. According to the Theorem of Bernstein-Gelfand we may interpret for generic τ the maps x h (ν1,... ,νn)(τ) Homg M(x τ),T(νn) T(ν1)M(x τ) ∈ · ··· · =Hom (χ(τ )) Id,T(νn) T(ν1) ∼ M → ··· as natural transformations of functors. Set now x h1 := h (ν + µ, µ, ν)(τ) Hom (χ(τ )) Id,T( ν)T( µ)T(ν+µ) , − − ∈ M → − − x h2 := h ( ν, ν)(τ + ν) Hom (χ(τ +ν)) Id ,T(ν)T( ν) , − ∈ M → − x h3 := h ( µ, µ)(τ + ν + µ) Hom (χ(τ +ν+µ)) Id,T(µ)T( µ) − ∈ M → −

and consider the natural transformations /

T (µ)T (ν)Gx(τ) / T (ν + µ)

T (µ)T (ν)h1 (h3)− 1

(T (µ)h2)− / T (µ)T (ν)T ( ν)T ( µ)T (ν + µ) / T (µ)T ( µ)T (ν + µ) − − − 1 1 x The diagram commutes, since (h3)− (T (µ)h2)− T (µ)T (ν)h1 as well as G (τ) imply the identity on M(x (τ + ν + µ◦)) under the◦ isomorphism · Hom (χ(τ )) T (µ) T (ν),T(ν+µ) M → ◦ ∼ Homg T (µ)T (ν)M(x τ),T(ν+µ)M(x τ) −→ · · ∼ Homg M(x (τ + ν + µ)),M(x (τ +ν+µ)) . −→ · · x Lemma 14. Set D (τ):=δx(ν+µ)(x τ)/δxν(x τ)δxµ(x (τ + ν)). Then the map x x · · · D G is algebraic on and there exists an algebraic extension on h∗ whose set of zeros has codimensionU 2. ≥ x x Proof. Since the maps d (ν1,... ,νn)h (ν1,... ,νn) have such algebraic extensions (Lemma 13), the commutativity of the above diagram implies that also DxGx has such an algebraic extension, where Dx(τ)=dx(ν+µ, µ, ν)(τ)dx( ν, ν)(τ + ν)dx( µ, µ)(τ + ν + µ) − − − − = δ (x τ)/δ (x τ)δ (x (τ + ν)). x(ν+µ) · xν · xµ · 246 KAREN GUNZL¨

Finally, we come to the following:

Proof of Theorem 3. By Lemma 11 it suﬃces to show that ∆(µ, ν; x)(τ)= cπ(τ+ρ) for a non-vanishing constant c. Note, that (Dx/De)(τ)= π(τ+ρ). We deduce from Lemma 12 that for generic weights DxDeGe =∆(µ,± ν; x)DeDxGx. e e x x Since D G as well as D G have algebraic extensions on h∗ which vanish only on a set of codimension 2 (Lemma 14), it follows that there is a constant c k×, independent of τ, µ, ν≥and x, such that ∈ c ∆(µ, ν; x)(τ)=(Dx/De)(τ)= π(τ+ρ). ±

7. Outlook 7.1. Identities. There are many nice identities for the triangle functions. Obvious are the Normalization identities. ∆(µ, ν; e)=1 ∆(0,ν;x)=1 ∆(ν, 0; x)=1. By means of Theorem 3 or directly with the deﬁnition of ∆ one then checks for ν, µ, η P and x, y : ∈ ∈W Decomposition identity. ∆(η + µ, ν; x)(τ)∆(η, µ; x)(τ + ν)=∆(η, µ + ν; x)(τ)∆(µ, ν; x)(τ). Rotation identity. 1 ∆(yµ,yν;x)(y τ)=(∆(µ, ν; y)(τ))− ∆(µ, ν; xy)(τ). · Flat triangle identity. Let ν, µ P be in the closure of a Weyl chamber. Then ∆(µ, ν; x)=1for all x . ∈ ∈W 7.2. Generalizations.

7.2.1. The Weyl group parameter. Let ν, µ P and x, y . Instead of applying the translation functors to the Verma modules∈ M(τ)and∈WM(x τ), we may choose the Verma modules M(x τ)andM(y τ). We then deﬁne a generalized· triangle · · function ∆g by 1 1 ∆g(µ, ν; y,x)(τ):=det yx− nat(µ, ν; x)(τ) (nat(µ, ν; y)(τ))− . ◦ ◦ We now have ∆g (µ, ν; e, x)(τ)=∆( µ, ν; x)(τ) and going back to the deﬁnition of ∆g we deduce the identities 1 (I1) ∆g(µ, ν; y,x)(τ)=(∆(µ, ν; y)(τ))− ∆(µ, ν; x)(τ), 1 (I2) ∆g(µ, ν; y,x)(τ)=(∆g(µ, ν; x, y)(τ))− and

(I3) ∆g(µ, ν; y,x)∆g(µ, ν; x, z)=∆g(µ, ν; y,z). The equivalent statement to the Rotation identity is obtained by comparing (I1) with 1 (I4) ∆g(µ, ν; y,x)(τ)=∆(yµ,yν;xy− )(y τ). · THE FINE STRUCTURE OF TRANSLATION FUNCTORS 247

7.2.2. Number of translations. The triangle functions measure in a subtle way the τ+ν+µ τ+ν τ+ν+µ relation between the two translation functors Tτ+ν Tτ and Tτ . Therefore

the triangle ◦

ν = µ

/ τ / ν+µ Let now integral weights ν1,... ,νn and µ1,... ,µm P be given such that n m ∈ i=1 νi = j=1 µj =: p.ThencallT(νi) the translation functor P P T (νi): ∞ χ(τ+ν1+ +νi 1) ∞ χ ( τ + ν 1 + +νi) M ··· − −→ M ··· M prχ(τ+ ν + +ν )(E(νi) M) 7→ 1 ··· i ⊗ and deﬁne similarly the translation functor T (µi). We now want to compare the functors T (νn) T(ν1)andT(µm) T(µ1) with each other and instead of

a triangle of translations◦···◦ we thus have now◦···◦ the following situation:

µ1 D µm

" τ τ + p

> <

> z

ν1 > νn

z / / We start by deﬁning a map nat(¯ µm,... ,µ1;νn,... ,ν1;x)(τ)forx and generic weight τ analogously to the deﬁnition of nat(µ, ν; x)(τ) as the composition∈W

Hom (χ(τ )) T (µm) T(µ1),T(νn) T(ν1) M → ◦···◦ ◦···◦ ∼ Homg T (µm) T(µ1)M(x τ),T(νn) T(ν1)M(x τ) −→ ··· · ··· · ∼ ˆ ˆ ˆ ˆ ˆ ˆ Homg Eµm Eµ1 M(x (τ +p)),Eνn Eν1 M (x (τ +p)) −→ ⊗···⊗ ⊗ · ⊗···⊗ ⊗ · ∼ E ∗ E∗ Eνn Eν1. −→ µ m ⊗···⊗ µ1 ⊗ ⊗···⊗ Here, we wrote Eµ for E(µ)xµ. We then obtain a generalized triangle function 3 by

1 3(µ ,...µ1;ν ,...ν1;x)(τ):=det x− nat(¯ µ ,...µ1;ν ,...ν1;x)(τ) m n ◦ m n 1 (nat(¯ µ ,...µ1;ν ,...ν1;e)(τ))− . ◦ m n Obviously we have 3(µ, ν; ν + µ; x)=∆(µ, ν; x)aswellas 1 3(µm,... ,µ1;νn,... ,ν1;x)(τ)=(3(νn,... ,ν1;µm,... ,µ1;x)(τ))− and we can reduce the calculation of 3 to the calculation of ∆ by means of the Split identities.

(1) 3(ν3,ν2,ν1;ν3 +ν2 +ν1;x)(τ)=∆(ν2,ν1;x)(τ)∆(ν3,ν1 +ν2;x)(τ)

=∆(ν2 +ν3,ν1;x)(τ)∆(ν3,ν2;x)(τ + ν1),

1 (2) 3(µ2,µ1;ν2,ν1;x)(τ)=∆(µ2,µ1;x)(τ) ∆(ν2,ν1;x)(τ) − . 248 KAREN GUNZL¨

ν2

/ O O

The ﬁrst identity just means that we can split ν1 ν3 into tri-

/ τ / ν1+ν2+ν3

ν2 ν2

/ /

8 N

O O p

p N

N p

ν ν ν N ν

1 p 3 1 3

p p

angles according to or N .Thisis

/ / τ / τ ν1+ν2+ν3 ν1+ν2+ν3

just the Decomposition identity in 7.1. The second equation describes the decom-

? ? =

= =

µ1 = µ2 µ1 µ2

= =

/ /

position of τ in τ / @ @ ?

? ?

ν1 ? ν2 ν1 ν2

? ?

Inductively, we may thus ﬁrst split up our situation into

µ1 D µm

/ / τ / τ + p

> < <

> p

> z

ν1 > νn

z / / In formulae:

3(µm,...µ1;νn,...ν1;x)(τ)=3(µm,...µ1;p;x)(τ) 3(p; νn,...ν1;x)(τ) 1 = 3(µm,...µ1;p;x)(τ) 3(νn,...ν1;p;x)(τ) − and in order to calculate the 3-functions it suﬃces thus to know them in the special m case 3(µm,... ,µ1; j=1 µj; x). This situation can then be reduced to triangles by decomposing it into

P µm 1

/ /

/ −

h D

h µ

µ1 h m

h D

h h h µ + +µ

h 1 m

h / τ ··· / τ + p or into

µm 1

/ /

/ −

D W

W D

µ W µ

1 W m

W +

µ1+ +µm + / τ ··· / τ + p Inductively one can then prove THE FINE STRUCTURE OF TRANSLATION FUNCTORS 249

Proposition. Let µ1 ,... ,µm P be integral weights and x .Then ∈ m ∈W m k 1 3(µm,... ,µ1; j=1 µj; x)(τ)= ∆(µk, i=1− µi; x)(τ) =2 kY P m 1 P − m k 1 = ∆( j=k+1 µj,µk;x)(τ + j=1− µj). k=1 Y P P Acknowledgment I am grateful to my advisor, Wolfgang Soergel, for his constant encouragement and all the fruitful discussions we had. Without his many ideas this work would not have been possible. I also want to thank Bernard Leclerc for bringing the article [Ka] to my attention. References

[AJS] Andersen, H.H., Jantzen, J.-C., Soergel, W.: Representations of quantum groups at a pth root of unity and of semisimple groups in characteristic p: Independence of p. Astérisque 220 (1994), 1–320. MR 95j:20036 [Be] Bernstein, J.N.: Trace in Categories. In Operator Algebras, Unitary Representations, En- veloping Algebras and Invariant Theory (A. Connes, M. Duflo, A. Joseph, R. Rentschler, editors), Actes du colloque en l’honneur de Jacques Dixmier, Progress in Mathematics 92 Birkhäuser, 1990, 417–423. MR 92d:17010 [BG] Bernstein, J.N., Gelfand, S.I: Tensor products of finite and infinite dimensional representations of semisimple Lie algebras. Compositio Math. 41 (1980), 245–285. MR 82c:17003 [BGG1] Bernstein, J.N., Gelfand, I.M., Gelfand, S.I.: Structure of representations generated by vectors of highest weight. Funct. Anal. App. 5 (1971), 1–8. MR 45:298 [BGG2] , Category of g-modules. Funct. Anal. App. 10 (1976), 87–92. [Di] Dixmier, J.: Enveloping Algebras. (North Holland Mathematical Library, Vol. 14) Amsterdam-New York-Oxford: North Holland, 1977. MR 58:16803b [ES] Etingof, P., Styrkas, K.: Algebraic integrability of Schrödinger operators and representations of Lie algebras. Compositio Math. 98 (1995), 91–112. MR 96j:58076 [EV] Etingof, P., Varchenko, A.: Exchange Dynamical Quantum Groups. On the xxx-preprint server under math.QA/9801135. [HS] Hilton, J.P., Stammbach, U.: A course in homological algebra. New York-Heidelberg- Berlin: Springer, 1971. MR 49:10751 [Ja1] Jantzen, J.C.: Moduln mit einem höchsten Gewicht. (Lecture Notes in Mathematics, Vol. 750) Berlin-Heidelberg-New York: Springer 1979. MR 81m:17011 [Ja2] , Einhüllende Algebren halbeinfacher Lie-Algebren. Berlin-Heidelberg-New York- Tokyo: Springer 1983. MR 86c:17011 [Ka] Kashiwara, M.: The Universal Verma Module and the b-Function. In R. Hotta, editor, Al- gebraic Groups and Related Topics, Proceedings of Symposia Kyoto 1983, Nagoya, 1983, Adv. Stud. in Pure Mathematics 6, Kinokuniya Tokyo and North-Holland, Amsterdam, 1985, 67–81. MR 87i:22047 [Ko] Kostant, B.: On the Tensor Product of a Finite and an Infinite Dimensional Represen- tation. Journal of Functional Analysis, 20 (1975), 257–285. MR 54:2888 [So] Soergel, W.: Kategorie , Perverse Garben und Modulnuber ¨ den Koinvarianten zur O Weylgruppe. Journal of the American Mathematical Society, Vol. 3, No. 2. April, 1990. MR 91e:17007

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