JOURNAL OF ALGEBRA 184, 839᎐851Ž. 1996 ARTICLE NO. 0287
Supercuspidal Representations of Finite Reductive Groups
Gerhard Hiss
IWR der Uni¨ersitat¨ Heidelberg, Im Neuenheimer Feld 368, D-69120 Heidelberg, Germany
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1. INTRODUCTION
A powerful tool in representation theory of finite groups of Lie type is Harish᎐Chandra induction. It is a method to construct, in a systematic way, representations for the group from those ofŽ. split Levi subgroups. Let G be a finite group of Lie type. An irreducible representationŽ of finite degree over a field containing the < 839 0021-8693r96 $18.00 Copyright ᮊ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. 840 GERHARD HISS tion is liftable to characteristic 0. This is related to a conjecture of Geck. We show that an irreducible representation is supercuspidal if and only if every composition factor of its projective cover is cuspidal. We then determine all supercuspidal unipotent representations of the classical groupsŽ. in odd characteristics and also of some exceptional groups of Lie type. The results indicate that a complete classification of all supercuspidal representations seems to be possible. 2. SUPERCUSPIDAL MODULES Let G be a finite group with a split BN-pair of characteristic p. Let L denote the set of all conjugates of standard Levi subgroups of G. Let k be a field which is a splitting field for every subgroup of G.We assume that the characteristic l of k is different from p. All modules G considered will be finitely generated. For L g L we denote by RL the functor Harish᎐Chandra induction, mapping finitely generated kL-modules to finitely generated kG-modules. The adjoint functor Harish᎐Chandra U G restriction or truncation is denoted by RL . We refer the reader tow 7, Sect. 1x for a general discussion of these functors. Harish᎐Chandra induction and restriction preserve projective modules. XX DEFINITION 2.1Ž Vigneraswx 29.Ž . For L, M .Ž, L, M .in the set W [ Ä4Ž.L, MLgL,Ma simple kL-moduleŽ up to isomorphism. , XX XX define ŽL, M .ŽF L, M .if and only if L F L and M is isomorphic to a U L XX XŽ. Ž .Ž . composition factor of RML . Define L, M 1 L, M if and only if X L X X Ž . L F L and M is isomorphic to a composition factor of RML . It is then easy to check that the relations F and 1 are partial orderings on W .If Ž.L,Mis a minimal element of W with respect to 1 , then M is called a supercuspidal simple kL-module. Remark 2.2. A kG-module M is called cuspidalŽ cf.wx 7, Definition 2.5 . if U L RLX Ž.M s 0 X X for all L g L with L strictly contained in L. It is easy to see that a kG-module is cuspidal if and only if each of its composition factors is. If M is a simple kL-module for some L g L , then M is cuspidal if and only if Ž.L,Mis a minimal element of W with respect to F . PROPOSITION 2.3. Let M be a simple kG-module. Then M is supercuspidal if and only if its projecti¨eco¨er PM is cuspidal. SUPERCUSPIDAL REPRESENTATIONS 841 Proof. We have G U G Hom kGŽ.Ž.P M, RV L Ž.(Hom kLRP LŽ. M ,V Ž.1 for every Levi subgroup L and every kL-module V. Now M is supercuspi- dal if and only if the left-hand side ofŽ. 1 is zero for every L g L properly contained in G and every kL-module V. On the other hand, PM is cuspidal if and only if the right-hand side ofŽ. 1 is zero for every such pair Ž.L, V . Remarks 2.4.Ž. a It follows from Proposition 2.3 that a supercuspidal module is cuspidal. Examples below show that the converse is not true in general. Ž.b Proposition 2.3 can be used to extend the definition of supercuspi- dal modules to arbitrary finitely generated kG-modules M. We say that M is supercuspidal if and only if PM is cuspidal. This is the case if and only if every composition factor of MrradŽ.M is supercuspidal. Ž.c Let G12and G denote groups with split BN-pairs of characteristic Ž p. Let Miidenote simple kM -modules, i s 1, 2. Then the simple kG1= . G212-module M m M is supercuspidal, if and only if each Miis. Ž.d Let MU denote the module contragradientŽ. dual to the kG-mod- ule M.If Mis simple, then M is supercuspidal if and only if MU is. 1 Ž.e Let M, V be simple kG-modules with Ext kGŽ.M, V / 0. If one of M or V is supercuspidal, then the other one is cuspidal. Ž.f If a cuspidal simple kG-module is in a block of defect zero, then it is supercuspidal. QUESTION 2.5. Let V be a simple kG-module. Then there is a minimal elementŽ.Ž.Ž. L, M 1 G, V . Is L, M uniquely determined Ž up to conjuga- tion. by V ? The corresponding question for cuspidal simple modules has an affirma- tive answerŽ seewx 22, Theorem 5.5. . 3. A CONJECTURE FOR FINITE REDUCTIVE GROUPS Let G be a connected reductive algebraic group which is defined over the finite field ކq with q elements. Let F denote the Frobenius endomor- F phism corresponding to this ކq-structure on G. The set G s G of fixed points of F on G is called a finite reductive group or a finite group of Lie type. It is a group with a split BN-pair of characteristic p dividing q. We choose a prime l different from p and a splitting l-modular system Ž.ŽK,R,kfor G where K is a finite extension of ޑl .. Let s be an F-stable semisimple element in the dual group GU. Its GU F-conjugacy class is 842 GERHARD HISS denoted by wxs , and the corresponding rational Lusztig series of irreducible K-characters of G by EŽG, wxs .Žseewx 5, p. 136 for the definition. . Now suppose that s is also l-regular, i.e., its order is not divisible by l. Then, by a result of Broue´ and Michelwx 2 , ElŽ.G,wxs [ E ŽG,wts x . DF tgCsGUŽ.t F is a union of l-blocks. Here, CsGUŽ.l is the set of F-stable l-elements in GU centralizing s. Conjecture 3.1. Let G be a finite reductive group and M a supercuspi- dal simple kG-module contained in ElŽG, wxs .. Then M is liftable to an RG-lattice whose character is contained in EŽG, wxs .. It is interesting to compare Conjecture 3.1 with a conjecture of Geck. Conjecture 3.2Ž Geckw 15,Ž. 6.6x. . The cuspidal simple KG-modules in the rational Lusztig series EŽG, wxs .remain simple on reduction modulo l. PROPOSITION 3.3. The truth of Geck’s Conjecture 3.2 implies the truth of Conjecture 3.1. Proof. Let M denote a supercuspidal simple kG-module. Then, by Proposition 2.3, the projective module PM is cuspidal. In particular, all ordinary constituents ofŽ. the lift of PM are cuspidal. Now M and hence also PMllie in a union of blocks E ŽG, wxs .. Byw 20, Theorem 3.1x there is a simple KG-module X contained in EŽG, wxs .such that M is a composition factor in a reduction modulo l of X. By Brauer reciprocity, X is a constituent of the lift of PM . Hence X is cuspidal. If Geck’s conjecture is true, the reduction modulo l of X is simple, hence isomorphic to M. In other words, M is liftable. Geck’s conjecture is known to be true for the groups GL nŽ.q w6, Theorem 3.5x and GUnŽ.q wx17, Theorem 6.10 , as well as for split classical groups and l s 2 Ž.q largewx 18, Theorem 4.4 . It is also true for some 3 exceptional groups such as Gq24Ž.wx19, 25, 24 and DqŽ.wx14 . Furthermore, it is known to be true in some cases for the unipotent cuspidal characters of Eq64Ž.wx16 and for FqŽ.wx30 . 4. SUPERCUSPIDAL UNIPOTENT MODULES OF CLASSICAL GROUPS We propose to classify all supercuspidal simple modules of the finite groups of Lie type. For this purpose the following criterion for a module for not being supercuspidal is very useful. SUPERCUSPIDAL REPRESENTATIONS 843 F Let G s G be an arbitrary finite group of Lie type. We preserve the notation of Sections 2 and 3. If is a K-valued class function of G, then ˆ denotes the restriction of to the set GlX of l-regular elements of G.If is a K-character of G, we view ˆ as Brauer character. The usual inner product on the set of K-valued class functions of G is denoted by ²:ᎏ, ᎏ . G If L is an F-stable Levi subgroup of G, we write RL for Lusztig’s F induction map from the set of class functions of L s L to the set of class functions of G. For a positive integer d we let ⌽d denote the dth cyclotomic polyno- mial. LEMMA 4.1. Let , , and denote ordinary characters of GŽ proper but not necessarily irreducible., such that is cuspidal, but no constituent of y is cuspidal. If ˆqˆˆs, then no modular constituent of ˆ is supercuspidal. Proof. Suppose that is a supercuspidal modular constituent of ˆ. Let ⌽ denote the character of the projective indecomposable module corre- sponding to . We then obtain a contradiction from the orthogonality relations 1 0 - Ý⌽Ž.g Ž.gy1 < 4.1. General Linear Groups In this paragraph we determine all supercuspidal simple kG-modules for Ž. GsGL n q . This result has independently been obtained by Vigneras wx29 . 844 GERHARD HISS It has been proved by Dipper and JamesŽ seewxwx 6, Theorem 3.5 and 8 . that every cuspidal simple kG-module is liftableŽ to an ordinary cuspidal module. and that every ordinary cuspidal module remains simple on reduction modulo l. Hence Conjecture 3.1 is true in this case. In order to determine the supercuspidal kG-modules, we have to find the ordinary simple cuspidal KG-modules first. The simple KG-modules are parameter- ized by pairs Ž.s, , where s is a semisimple element in G and is a simple unipotent KCGsŽ. s -module. Let X ,denote the simple KG-module corre- sponding to the pair Ž.s, . Then Xs, is cuspidal if and only if s is a regular element in the Coxeter torus of G. In this case is the trivial module, of course. THEOREM 4.2. Let M be a cuspidal kG-module and let Xs,1 be a lift of M Ž.i.e., s is a regular element in the Coxeter torus of G . Then M is supercuspidal if and only if the lX-part of s is regular. XX XX Proof. Let s denote the l -part of s. Then Xs,1 lies in ElŽG, ws x..If s X is not regular, EŽG, ws x.Žcontains no cuspidal KG-module seewx 28, 7.5.4. . This implies that M is not supercuspidal, since bywx 20, Theorem 3.1 , PM X has at least one ordinary constituent in EŽG, w s x.. X X On the other hand, if s is regular, all ordinary modules in ElŽG, w s x. are cuspidal, and thus M is supercuspidal. 4.2. Unitary Groups Ž. In this paragraph, G s GUn q is the general unitary group of degree n 2 over the field with q elements. We use the notation ofwx 10 . For & n,a partition of n, let denote the corresponding irreducible unipotent character of G. It is known that is the character of a cuspidal representation if and only if n s aaŽ.q1r2 for some integer a G 1 and is the triangular partition of n. Ž. Ž. THEOREM 4.3. Let n s aaq1r2and G s GUn q . Denote by X the cuspidal unipotent KG-module and by M its reduction modulo lŽ which is irreducible by wx17. . Let d denote the multiplicati¨e order of yq modulo l. Then M is supercuspidal if and only if d is e¨en or d G 2 a y 1. Proof. If d is even or if d ) 2 a y 1, then X is of l-defect zeroŽ see the formula for the dimension of X inw 28,Ž. 9.5.1x. and so M is supercuspidal ŽŽ.Remark 2.4 f. . If d s 2 a y 1, the defect group of the l-block of G containing M is cyclic bywx 10 . By the results of Fong and Srinivasanw 13, Ž.6Ax on the Brauer trees of G, the node corresponding to X on the Brauer tree is joined to the exceptional node, which itself corresponds to cuspidal KG-modules. By Proposition 2.3, M is supercuspidal.Ž Note that the results ofwx 13 have only been proved for odd q, but the proof given there carries over to the case of even q.. SUPERCUSPIDAL REPRESENTATIONS 845 Suppose now that d is odd and smaller than 2 a y 1. Let denote the triangular partition corresponding to the cuspidal module X. Consider the Ž. Ž. subgroup G01= G of G with G0( GUnydq and G1( GUdq . Let Žd. L11(GU q F G 1and put L s G011= L . Let be a partition of n y d obtained from by removing a d-hook, and let denote the irreducible 1 character of G01corresponding to . Since is obtained from 1by adding a d-hook, we have ²: , RG 1 "12Ž. LŽ.1m s byw 10,Ž.Ž. 2.12 , 2Dx . Ž. Let t g L1 be an l-element such that CtG sL. The existence of such X an element may be seen as follows. Embed L11in a Coxeter torus L of 2 2m GL ddŽq .. Since d is odd, l<⌽ Žq .. Suppose that l is the exact power of l 2d XX dividing q 1. Bywx 17, Lemma 7.3 , there is an l-element t L1 which y d g m Ž X .q y1 has d distinct eigenvalues of order l . Then t [ t g L1 has the desired properties. It follows that there is a character of L1 of l-power order such that [ RG GL LŽ.1m is an irreducible character of G. Since 1 is not a triangular partition, is not cuspidalŽ seew 28,Ž. 7.8.2x. . Then ˆ is a Brauer character of G and, by general properties of the Lusztig functor, we have RˆG RˆG 1. ˆs GL LŽ.11m sGL L Ž.m Ž. By 2 , this can be written as ˆ q ˆˆs , where and are ordinary characters and y contains no cuspidal constituent. By Lemma 4.1, ˆ is not supercuspidal. 4.3. Symplectic and Orthogonal Groups In this paragraph we determine the supercuspidal unipotent characters for the symplectic and orthogonal groups. Since we are dealing only with unipotent characters, we do not have to worry about the isogeny type of Ž. Ž. the group. We let G be one of the groups Spnnq , n G 4 even, SO q , "Ž. nG7 odd, or SOn q , n G 8 even. Thus the type of G is one of B, C, D, or 2D, respectively. Define the parameter to be 0 if G is of type B or C; otherwise let s 1. If G has a cuspidal unipotent character, it has exactly one; we denote it by ⌬ in the following. In case G is of type Bmmor C , there is a cuspidal 2 unipotent character if and only if m is of the form m s a q a for some 2 a G 1. In case G is of type Dmmor D , there is a cuspidal unipotent 846 GERHARD HISS 2 character if and only if m is of the form m s a for some even, respec- tively, odd a G 1. The cuspidal character corresponds to the symbol ⌬ 0,1,...,2a y [ ž/ᎏ. Let l be an odd prime not dividing q and let e denote the order of q 2 modulo l. Following Fong and Srinivasanwx 12 , we say that l is a linear e e prime if l divides q y 1 and a unitary prime if l divides q q 1. THEOREM 4.4. Let the notation be as abo¨e. Suppose that G has a cuspidal unipotent KG-module X. Let M denote a reduction of X modulo l. Then M is irreducible and supercuspidal if l is a linear prime or if e G 2 a y . If l is a unitary prime and e - 2 a y , then no composition factor of M is supercuspidal. Proof. The character of X is ⌬. From the degree formulaw 28, Theo- rem 8.2x one can conclude that ⌬ is of l-defect 0 if l is linear or if e ) 2 a y . We may therefore assume that l is a unitary prime. If e s 2 a y , then ⌬ lies in a block with a cyclic defect group and is an end node of the Brauer treew 13,Ž. 6Ax . In other words, the restriction of ⌬ to the l-regular classes is an irreducible Brauer character. Since ⌬ is connected to the exceptional node and since the exceptional characters are cuspidal in this case, it follows that ˆ⌬ is supercuspidalŽ again also true for even q, although only proved for odd q inwx 13. . We now show that M has no supercuspidal composition factors for e - 2 a y . Let L s G01= L denote a maximal e-split Levi subgroup of e Ž G, where L1 is a cyclic torus of order q q 1. Seewx 1, p. 52 for a discussion of the e-split Levi subgroups of G.. Consider the symbol 0,1,...,2a 1 ⌬ [ y y . 1 ž/2ayey Let ⌬ correspond to the unipotent character of G . Then is not 1 ⌬110 ⌬ cuspidal. Byw 11,Ž. 3.2x , we have ²: , RG 1 / 0. ⌬LŽ.⌬1m Let GU denote the finite reductive group dual to G. We may embed GU U into GL nŽ.q in such a way that L11, a torus dual to L , embeds into a U Coxeter torus of GL 2 eŽ.q . Since l<⌽2 eŽ.q , there is an l-element in L1 whose eigenvalues are all distinct and different from 1wx 17, Lemma 7.3 . It U UŽ. follows that CtG sL. By duality, there is an irreducible K-character of L such that RGŽ. is an irreducible character of G.We 1 GL L ⌬1m can now complete the proof as in the case of the unitary groups. SUPERCUSPIDAL REPRESENTATIONS 847 We believe that G has no unipotent 2-modular supercuspidal represen- tations. 4.4. Some Exceptional Groups In the final paragraph of this section we collect some results about the supercuspidal unipotent representations of exceptional groups of Lie type. For this purpose let G be a simple, simply connected algebraic group of one of the types D426, G , E ,orE 7. Furthermore, F is a Frobenius endomorphism of G with respect to some ކq-rational structure on G such Ž. F 3 Ž. Ž. that the finite group G s Gq sG is equal to one of Dq42,Gq, 2 Ž.Ž.Eq6 ad ,or Ž.Ž.ŽEq7 ad . We do not consider Fq46Ž.,EqŽ.,orEq 8 Ž., since in these cases we have only very little knowledge about the supercuspidal unipotent representations.. Let l be a prime not dividing q. We write e for the multiplicative order of q modulo l. Ž. 3 Ž. THEOREM 4.5. 1 Let G s Dq4 .Then both cuspidal unipotent repre- 3 3 sentations D44wxy1 and D wx1 remain irreducible modulo l. Furthermore we ha¨e: $$ Ž. 33 aIf l s 2, then D44wxy1 and D wx1 are not supercuspidal. $ Ž. 3 Ä4 bIf l ) 2, then D4 wxy1 is supercuspidal if and only if e f 2, 6 . 3 Ž.c Suppose that l ) 2 and that D wx1 is not of defect 0. Then $4 Ä4 3 eg2, 3, 6 or e s 1 and l s 3. Also, D4 wx1 is supercuspidal if and only if es6and theŽ. unknown decomposition number cŽ in the table for the case 2 lq< yqq1in wx14, p. 3264 .is 0. Ž. Ž . 2 Let G s Gq2 .Then all four cuspidal unipotent representations G2wx1, 2 G22wxy1, G wx , and G 2 w xremain irreducible modulo l. Furthermore we ha¨e: Ž. $ a G2 wx1 is supercuspidal if and only if l ) 3 and e / 2. Ž. $ bG2 wxy1is supercuspidal if and only if l ) 2 and e / 2. 2 Ž.c G22wxand G w xare supercuspidal if and only if l / 3 and e / 3. Ž. Ž . Ž . 3 Let G s E6 ad q and suppose that l G 5. Then the following holds. 2 Ž.a The two cuspidal unipotent characters E66wx and E w xare not of defect 0 if and only if e g Ä43, 6, 9, 12 . Ž. 2 b If e s 12, then E66wx and E w xremain irreducible modulo l and gi¨e rise to supercuspidal l-modular representations. $ $ Ž. Ä4 2 c If e g 3, 6, 9 , then no constituent of E66wx or of E w xis supercuspidal. Ž. Ž . Ž . 4 Let G s E7 ad q and suppose that l G 7. Then the following holds. 848 GERHARD HISS Ž. a The two cuspidal unipotent characters E77wx and E wy xare not of defect 0 if and only if e g Ä42, 6, 10, 14, 18 . Ž. Ä4 b If e g 10, 14, 18 , then E77wx and E wy xlie in cyclic l-blocks and remain irreducible modulo l. Ž. Ä4 $$ c If e g 2, 6, 14 , then no constituent of E77wx or of E wy xis supercuspidal. In all of the abo¨e cases, the supercuspidal unipotent l-modular representa- tions of G are liftable. Proof. Ž.1 This follows from the decomposition matrices given inw 14, Proposition 5.3x and Proposition 2.3. For partŽ. c one also has to observe $ 3 that the projective indecomposable module corresponding to D4 wxy1 contains every noncuspidal ordinary simple module with multiplicity c. Ž.2 This follows from the decomposition matrices given inwx 19, 25, 24 and Proposition 2.3. Ž.3 The statement in Ž. a follows by inspecting character degrees, and Ž. b follows fromw 23, Theorem 3.1Ž. 8x . U Ž.Ž. Now suppose that e s 3. Let G s Eq6 sc denote the group dual to G, and consider the maximal torus T U of GU which is a direct product of 2 U three cyclic groups of order q q q q 1, i.e., T is the Sylow ⌽3-torus of GU. In each of the three cyclic factors of T U, choose an l-element of maximal order, and let t denote the product of these elements. Then U UŽ. CtG sT, as can easily be checked with the tables given inwx 9 . By duality, there is an irreducible K-character of T in general position, U GŽ. where T is a maximal torus of G dual to T . Thus [ GTR Ts GŽ. ² GŽ. : RT is an irreducible character of G. We have RT 1,E6wx s ² GŽ. 2:Ž RT 1,E6w x sy9. These multiplicities can be calculated from the character table of the Weyl group of type E6 and the Fourier transform matrices given by Lusztig. I am indebted to Gunter Malle for allowing me $ .ŽGG.Ž. to use his explicit tables. Since ˆs RˆˆTT s R 1 sy9E6 y $ wx 2ˆ 9E6 wx q, where is a ޚ-linear combination of unipotent characters of G, none of which is cuspidal, the result follows from Lemma 4.1. U Now let e s 6 and let t g G denote an l-element whose centralizer in UU2Ž. Ž2 . G is a Levi subgroup L of type Aq2 with central torus q q q q 1 = 2 U Žq yqq1.Ž seewx 9. . Let L denote a Levi subgroup of G dual to L . Put G [GLR LŽ.with an irreducible linear K-character of L correspond- ing to t by duality. Then is an irreducible character of G, which is not ² :² 2: cuspidal bywx 28, 7.5.4 . On the other hand, , E66wx s , E w xsy1, which can easily be calculated since every unipotent character of L is GŽ. uniform. Since ˆs GLRˆ L1 , the result follows from Lemma 4.1. The case e s 9 follows fromw 23, Theorem 3.1Ž. 7x . SUPERCUSPIDAL REPRESENTATIONS 849 Ž.4 The statement in Ž. a follows by inspecting character degrees. Now let egÄ410, 14, 18 . Then the Sylow l-subgroup of G is cyclic. The two charac- Ž ters E77wx and E wy xare complex conjugate to each other. They correspond to a pair of complex conjugate eigenvalues of the Frobenius map on a certain cohomology groupŽ seew 27,Ž. 7.3x. and thus are indeed complex conjugate.. All other characters in the l-block containing these are real valued. Hence E77wx and E wy xare connected to the real stem on the Brauer tree. This implies that they remain irreducible modulo l. PartŽ. c can be proved with similar methods as the corresponding part for Ž.Ž.Eq6ad . At the present we can prove neither that the reductions modulo l of Ž E77wxand E wy xare supercuspidal if e s 18 which would follow from a conjecture inwx 23. nor that they are not supercuspidal if e s 10. 2 In the case of Eq6Ž.we cannot, at the moment, exclude the possibility 2 2 Ž 2 2 2. that E66wx1 q E wx and E 66wx1 q E w x are characters of projective modules in characteristic l such that e s 3. This possibility would give rise to a supercuspidal l-modular representation which is not liftable. Of course, this would also be a counterexample to Geck’s Conjecture 3.2. We do not believe that this can happen; these remarks are only included to give the reader some idea about the remaining problems. Let us finally comment on the supercuspidal representations for the 2 Suzuki groups and the Ree groups of type Gq2Ž.. These groups are not finite reductive groups as introduced in Section 3, but they are groups with split BN-pairs. Thus it makes sense to talk about their supercuspidal representations. 2 Ž. 22nq1 THEOREM 4.6. Let G s Bq2 ,qs2,nG0be a Suzuki group, 2 and let l be an odd prime. Then both cuspidal unipotent representations B2wx a 2 and Bwx b remain irreducible modulo l. In particular all supercuspidal 2 $$ 22 unipotent representations of G are liftable. Furthermore, B22wx a and B wx b 2 are supercuspidal if and only if l ¦ q q '2 q q 1. Ž. 2 Ž. 22nq1 2Let G s Gq2 ,qs3,nG0be a Ree group. Let i denote the ith entry of the table of unipotent representations of G gi¨en in w4, pp. 488᎐489x . Then 34, ,..., 8are the six cuspidal unipotent representations of G.Let l / 3 be a prime. If l is odd, then 34, ,..., 8remain irreducible modulo l. Furthermore we ha¨e: Ž. a 34and ha¨e the same reduction modulo 2 which is irreducible and supercuspidal. There are no other unipotent supercuspidal 2-modular representations. Ž.b If l is odd, ˆˆˆ345, , , and ˆ 6are supercuspidal if and only if 2 l¦qq'3qq1. 850 GERHARD HISS Ž. ˆˆ 2 c If l is odd, 78and are supercuspidal if and only if l ¦ q q 1. In particular all supercuspidal unipotent representations of G are liftable. Proof. This can easily be proved with the results inwx 3, 26, 21 . We leave the details to the reader. For partŽ.Ž. 2 a use the fact that 34and lie in a 2-block with a cyclic defect group of order 2. 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