Weil Representations of Finite General Linear Groups and Finite Special Linear Groups

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Paciﬁc Journal of Mathematics

WEIL REPRESENTATIONS OF FINITE GENERAL LINEAR GROUPS AND FINITE SPECIAL LINEAR GROUPS

PHAM HUU TIEP

Volume 279 No. 1-2 December 2015

PACIFIC JOURNAL OF MATHEMATICS Vol. 279, No. 1-2, 2015 dx.doi.org/10.2140/pjm.2015.279.481

WEIL REPRESENTATIONS OF FINITE GENERAL LINEAR GROUPS AND FINITE SPECIAL LINEAR GROUPS

PHAM HUU TIEP

Dedicated to the memory of Professor R. Steinberg

Let q be a prime power and let Gn be the general linear group GLn.q/ or the special linear group SLn.q/, with n 4. We prove two characterization theorems for the Weil representations of Gn in any characteristic coprime to q, one in terms of the restriction to a standard subgroup Gn 1, and another in terms of the restriction to a maximal parabolic subgroup of Gn.

1. Introduction

The so-called Weil representations were introduced by A. Weil[1964] for classical groups over local fields. Weil mentioned that the finite field case may be considered analogously. This was developed in detail by R. E. Howe[1973] and P. Gérardin [1977] for characteristic zero representations. The same representations, still in characteristic zero, were introduced independently by I. M. Isaacs[1973] and H. N. Ward[1972] for symplectic groups Sp2n.q/ with q odd, and by G. M. Seitz [1975] for unitary groups. (These representations for Sp2n.p/ were also constructed in[Bolt et al. 1961].) Weil representations of finite symplectic groups Sp2n.q/ with 2 q were constructed by R. M. Guralnick and the author in[Guralnick and Tiep j 2004]. Weil representations attract much attention because of their many interesting features; see, for instance, [Dummigan 1996; Dummigan and Tiep 1999; Gow 1989; Gross 1990; Scharlau and Tiep 1997; 1999; Tiep 1997a; 1997b, Zalesski 1988]. The construction of the Weil representations may be found in[Howe 1973; Gérardin 1977; Guralnick and Tiep 2004; Seitz 1975], etc. In particular, in the case of general and special linear groups, they can be constructed as follows. Let W ކn D q with n 3, and let ރ, respectively ކq, be a fixed primitive .q 1/-th root of Q 2 2 unity. Then SL.W / SLn.q/ has q 1 complex Weil representations, which are D The author gratefully acknowledges the support of the NSF (grant DMS-1201374) and the Simons Foundation Fellowship 305247. MSC2010: 20C15, 20C20, 20C33. Keywords: Weil representations, finite general linear groups, finite special linear groups.

481 482 PHAM HUU TIEP the nontrivial irreducible constituents of the permutation representation of SL.W / on W 0 . The characters of these representations are i , where 0 i q 2 and nf g n;q Ä Ä q 2 i 1 X ik .g k 1 / (1-1) .g/ qdim Ker W 2ı : n;q D q 1 Q 0;i k 0 D 2 Similarly, GL.W / GLn.q/ has .q 1/ complex Weil representations, which are the q 1 nontrivialD irreducible constituents of the permutation representation of GL.W / on W 0 , tensored with one of the q 1 representations of degree 1 n f g of GLn.q/. If we fix a character ˛ of order q 1 of GLn.q/, then the characters i;j of these representations are n;q, where 0 i; j q 2 and Ä Ä Â q 2 Ã i;j 1 X ij .g k 1 / j (1-2) .g/ qdim Ker W 2ı ˛ .g/: n;q D q 1 Q 0;i k 0 D i;j i Note that n;q restricts to n;q over SLn.q/. From now on, let us fix a prime p, a power q of p, and an algebraically closed field ކ of characteristic ` not equal to p. If G is a finite (general or special) linear group, unitary group, or symplectic group, then by a Weil representation of G over ކ, we mean any composition factor of degree > 1 of a reduction modulo ` of a complex Weil representation of G. As it turns out, another important feature of the Weil representations is that, with very few small exceptions, Weil representations are precisely the irreducible ކG-representations of the first few smallest degrees (larger than 1); see [Brundan and Kleshchev 2000; Guralnick et al. 2002; 2006, Guralnick and Tiep 1999, 2004, Hiss and Malle 2001; Tiep and Zalesski 1996]. Aside from this characterization by degree, Weil representations can also be recognized by various conditions imposed on their restrictions to standard subgroups, or parabolic subgroups. This was done in the case where G is a symplectic group or a unitary group, in[Tiep and Zalesski 1997] for complex representations and in [Guralnick et al. 2002, 2006, Guralnick and Tiep 2004] for modular representations (in cross characteristics). However, the case where G is a general or special linear group has not been treated. Perhaps one of the reasons for the absence until now of such characterizations (as regards the restriction to a parabolic subgroup) is that the obvious ana- logue of[Guralnick et al. 2002, Corollary 12.4] fails in this case; see Example 4.1. n We now fix W ކ with a basis .e1;:::; en/, and consider Gn G GL.W / D q D D or SL.W /. By a standard subgroup Gm in Gn, where 1 m n 1, we mean Ä Ä (any G-conjugate of) the subgroup Gm StabGn e1;:::; em ކq ; em 1;:::; en : D h i C Next, we fix a primitive p-th root of unity ރ. A maximal parabolic subgroup of G is conjugate to 2 P StabG e1;:::; e ކ D h k i q WEIL REPRESENTATIONS OF FINITE GENERAL LINEAR GROUPS 483 for some 1 k n 1. The unipotent radical Q of P is Ä Ä Â Ã ˇ Ik X ˇ ŒX ˇ X Mk;n k .ކq/ ; WD 0 In k ˇ 2 where M .ކq/ is the space of all a b matrices over ކq. Any irreducible Brauer a;b character of Q can then be written in the form

Trކ =ކ tr.XY / ˇY ŒX q p W 7! for a unique Y Mn k;k .ކq/. We deﬁne the rank of ˇY to be the rank of the 2 matrix Y . The main results of the paper are the following theorems, which characterize the Weil representations of ﬁnite general and special linear groups in terms of their restrictions to a standard or maximal parabolic subgroup.

Theorem A. Let q be a prime power, n 5, and for any integer m 4, let Gm denote the general linear group GLm.q/ or the special linear group SLm.q/. Con- sider the standard embedding of Gm in Gn when n > m. Let ކ be an algebraically closed ﬁeld of characteristic zero or characteristic coprime to q. Then for any ﬁnite-dimensional irreducible ކGn-representation ˆ, the following statements are equivalent:

(i) Either deg ˆ 1, or ˆ is a Weil representation of Gn. D (ii) ˆ has property .W/; that is, if ‰ is any composition factor of the restriction

ˆ Gn 1 , then either deg ‰ 1, or ‰ is a Weil representation of Gn 1. j D (iii) For some m with 4 m n 1, every composition factor of the restriction Ä Ä ˆ G either is a Weil representation or has degree 1. j m (iv) For every m with 4 m n 1, every composition factor of the restriction Ä Ä ˆ G either is a Weil representation or has degree 1. j m Theorem B. Let q be a prime power, n 4, and let G denote the general linear group GLn.q/ or the special linear group SLn.q/. Let ކ be an algebraically closed ﬁeld of characteristic zero or characteristic coprime to q. Let W ކn denote D q the natural G-module, and let Pk be the stabilizer of a k-dimensional subspace of W in G, where 2 k n 2. Then for any ﬁnite-dimensional irreducible Ä Ä ކGn-representation ˆ, the following statements are equivalent:

(i) Either deg ˆ 1, or ˆ is a Weil representation of Gn. D (ii) ˆ has property .P /; that is, the restriction ˆ Q to the unipotent radical Q of k j P contains only irreducible ކQ-representations of rank 1. k Ä Note that the deﬁnitions of properties .W/ and .Pk / do not depend on the choice of the particular standard or parabolic subgroup. Our subsequent proofs also make use of another local property .Z/, which is deﬁned by the condition (2-1) in Section 2. 484 PHAM HUU TIEP

Theorem B has already been used by A. E. Zalesski[ 2015] in his recent work. Throughout the paper, we will say that an ordinary or Brauer character ' of GLn.q/ or SLn.q/ has property .W/, or .Pk /, if a representation affording ' does as well. The notation IBr.X / denotes the set of irreducible Brauer characters of a ﬁnite group X in characteristic `. If Y is a subgroup of a ﬁnite group X , and ˆ is an ކX -representation and ‰ is an ކY -representation, then ˆ Y is the restriction of j ˆ to Y , and ‰Y is the ކX -representation induced from ‰, with similar notation for ordinary and Brauer characters, as well as for modules. If is a complex character of X , then ı denotes the restriction of to the set of `0-elements in X .

2. Local properties and Weil representations

A key ingredient of our inductive approach is the following statement:

Proposition 2.1. Let G Gn GLn.q/ with n 5 and let ˆ be an ކG-representation. D D Let K G4 be a standard subgroup of G. Then the following statements hold. D (i) If ˆ has property .P / for some 2 k n 2, then the ކK-representation k Ä Ä ˆ K has property .P2/. j (ii) If ˆ K has property .P2/, then ˆ has property .P / for all 2 k n 2. j k Ä Ä Proof. Consider P StabG e1;:::; ek ކq ; D h i K StabG e1; e2; ek 1; ek 2 ކq ; e3;:::; ek ; ek 3;:::; en : D h C C i C Then P2 P K plays the role of the second parabolic subgroup of K, the WD \ stabilizer in K of the plane e1; e2 ކ , with unipotent radical Q2 Q K. Let ' h i q WD \ denote the Brauer character of ˆ.

(i) Suppose that ˆ has property .P /. Consider any irreducible constituent ˇ ˇY k D of ' Q. By the assumption, rank.Y / 1. Writing j Ä ÂY Y Ã Y 1 2 ; D Y3 Y4 where Y1 is 2 2, it is easy to see that the restriction ˇ Q is just the character ˇY j 2 1 of Q2. Certainly, rank.Y1/ rank.Y /. It follows that ˆ K has property .P2/. Ä j (ii) Suppose that ˆ does not possess property .P / for some 2 k n 2. Then k Ä Ä we can ﬁnd an irreducible constituent ˇ ˇY of ' Q, where rank.Y / r 2. D j DW Note that the conjugation by the element diag.A; B/ in the Levi subgroup

L GLk .q/ GLn k .q/ D of P sends ˇY to ˇ 1 . Replacing ˇ by a suitable L-conjugate, we may assume B YA that the principal r r submatrix of Y is the identity matrix Ir . Now, in the notation WEIL REPRESENTATIONS OF FINITE GENERAL LINEAR GROUPS 485 of (i), we have ˇ Q ˇY , where Y1 I2. But this violates property .P2/ j 2 D 1 D for ˆ K . j Corollary 2.2. Let G Gn GLn.q/ with n 5 and let ˆ be an ކG-representation. D D Let H Gm be a standard subgroup of G for some 4 m n 1. Then the D Ä Ä following statements hold. (i) If ˆ has property .P / for some 2 k n 2, then the ކH -representation k Ä Ä ˆ H has property .Pj / for all 2 j m 2. j Ä Ä (ii) If ˆH has property .Pj / for some 2 j m 2, then ˆ has property .P / Ä Ä k for all 2 k n 2. Ä Ä Proof. Consider a standard subgroup K G4 of H . D (i) By applying Proposition 2.1(i) to ˆ, ˆ K has property .P2/. Hence, by applying j Proposition 2.1(ii) to ˆ H , ˆ H has property .Pj / for all 2 j m 2. j j Ä Ä (ii) By applying Proposition 2.1(i) to ˆ H , ˆ K has property .P2/. Hence, by j j applying Proposition 2.1(ii) to ˆ, ˆ has property .P / for all 2 k n 2. k Ä Ä We will also fix the following elements in GLn.q/: 01 1 01 Â1 1Ã Â1 1Ã Â1 1Ã x In 2; y In 4; z @0 1 1A In 3: D 0 1 ˚ D 0 1 ˚ 0 1 ˚ D ˚ 0 0 1 If ' is a Brauer character of G, we define 'Œ1 '.1/ .q 1/'.x/ q'.y/; 'Œ2 '.y/ '.z/: WD C C WD We will furthermore say that ' (or any representation affording it) has property .Z/ if (2-1) 'Œ1 'Œ2 0: D D Corollary 2.3. Let G GLn.q/ or SLn.q/ with n 4 and let ˆ be a Weil repre- D sentation of G over ކ. Then ˆ has property .P / for all 2 k n 2. If n 5, k Ä Ä then ˆ has property .W/. Proof. It suffices to prove the statement in the case where ކ ރ and furthermore i;j i D G GLn.q/, as n;q restricts to over SLn.q/. D n;q (i) First we consider the case G GL4.q/ and consider the parabolic subgroup D P StabG. e1; e2 ކ /, with unipotent radical Q. It is easy to check that IBr.Q/ D h i q consists of three P-orbits: 1Q , O1 of characters of rank 1, and O2 of characters f g of rank 2; moreover, 2 2 2 O1 .q 1/.q 1/; O2 .q 1/.q q/: j j D C j j D i;j i;j If q 3, then O2 > .1/ for all i; j , whence . / Q can afford only characters j j 4;q 4;q j of rank 1, and so we are done. Ä 486 PHAM HUU TIEP

Suppose that q 2, in which case there is only one complex Weil character 1;1 D of degree 14. We also consider the irreducible complex character of D 4;2 G GL4.2/ A8 of degree 7. As O1 9, O2 6, and Q Ker./, we must D Š j j D j j D 6Ä have that X Q 1Q : j D C O2 2 For the aforementioned involutions x, y, we have that

.x/ 6; .y/ 2; D D whence x belongs to class 2A and y belongs to class 2B in the notation of[Conway et al. 1985]. It follows that

.x/ 1; .y/ 3; D D and so X X X X .x/ 2; .y/ 2; .x/ 1; .x/ 3: D D D D O2 O2 O1 O1 2 2 2 2 Since X X Q a 1Q b c j D C C O1 O2 2 2 for some nonnegative integers a; b; c, we conclude that .a; b; c/ .5; 1; 0/, i.e., D has property .P2/, as desired.

(ii) Now we consider the general case of G GLn.q/ with n 5, and consider D a standard subgroup H GL4.q/ and a standard subgroup L GLn 1.q/ in G. D n D Let n denote the permutation character of G on ކq, so that

q 2 X i;0 n 2 1G: D n;q C i 0 D n 4 Note that .n/ L qn 1, and so ˆ has property .W/. Similarly, .n/ H q 4. j D j D Furthermore, according to (i), 4 has property .P2/. It follows that .n/ H also j has property .P2/, and so, by Corollary 2.2(ii), n has property .Pk /. Consequently, i;0 i;j n;q and n;q also possess property .Pk /.

Lemma 2.4. Let G GLn.q/ or SLn.q/, where n 4, and let ˆ be a Weil D representation of G over ކ. Then ˆ has property .Z/. Proof. Let ' be the Brauer character of ˆ. It is well known, see[Guralnick and Tiep 1999] for instance, that ' is a linear combination of the reduction modulo ` i;j i of some n;q or (note that such reductions need not be irreducible) and a D n;q WEIL REPRESENTATIONS OF FINITE GENERAL LINEAR GROUPS 487 linear character of G. As any linear character has property .Z/, it sufﬁces to show that has property .Z/. According to (1-2), we have n n 1 n 2 q 1 q 1 q 1 .1/ ı0;i; .x/ ı0;i; .y/ .z/ ı0;i; D q 1 D q 1 D D q 1 which implies property .Z/ for .

Proposition 2.5. Let G GLn.q/ and S SLn.q/ G with n 4. Let ˆ be an D D Ä ކG-representation and let ‰ be an ކS-representation. Also, let P .P / for some D k 2 k n 2, or P .W/. Ä Ä D (i) ˆ has property P if and only if ˆ S has property P. j (ii) ‰ has property P if and only if ‰G has property P. Proof. (a) First we consider the case P .P / and let P be the stabilizer in G D k of a k-space in the natural module W ކn, with unipotent radical Q. Note that D q Q < P S. Furthermore, if O1 denotes the set of all Brauer irreducible characters \ of Q of rank 1, then O1 forms a single P-orbit and also a single P S-orbit. \ It is clear that ˆ has property .Pk / if and only if ˆ S has property .Pk /, since j G Q < S. It is also clear that ‰ has property .Pk / whenever ‰ has property .Pk /, G since ‰ Q is a subquotient of .‰ / Q. Assume now that ‰ has property .P / j j k and affords the Brauer character . Then, by the aforementioned discussion, P Q a 1Q b for some integers a; b 0 and O1 . Note that we j D C WD 2 can ﬁnd a cyclic subgroup C Cq 1 of P such that G S Ì C , and again by the Š D aforementioned discussion, C preserves . As

G X c . / Q . / Q; j D j c C 2 G G we conclude that . / Q .q 1/.a 1Q b /, and so ‰ has property .P /. j D C k (b) Next we consider the case P .W/ and let H GLn 1.q/ be a standard D Š subgroup of G, so that H S SLn 1.q/ is also a standard subgroup of S. \ Š We already mentioned that Weil representations of H restrict irreducibly to Weil representations of H S. In particular, if ˆ has property .W/ then so does ˆ S . \ j Similarly, as the composition factors of ‰ are among the composition factors G G of .‰ / S , if ‰ has property .W/ then so does ‰. j Conversely, observe that q 2 X . i /G i;j : m;q D m;q j 0 D Hence, if ‚ is a Weil representation of SLm.q/ and m 3, then every composition GLm.q/ factor of ‚ is a Weil representation of GLm.q/ or a representation of degree 1. Moreover, we can ﬁnd a cyclic subgroup C Cq 1 of H such that G S ÌC and Š D 488 PHAM HUU TIEP

H .H S/ Ì C . It follows that if ‰ has property .W/ then so does ‰G. Finally, D \ G as the composition factors of ˆ are among the composition factors of .ˆ S / , G j if ˆ S has property .W/ then so do .ˆ S / and ˆ. j j Corollary 2.6. Let G GLn.q/ or SLn.q/ and let ˆ be an ކG-representation. D (i) If ˆ possesses property .W/ and n 5, then ˆ has property .Z/. (ii) If ˆ has property .P / for some 2 k n 2 and n 4, then the Brauer k Ä Ä character ' of ˆ satisfies 'Œ1 0. D (iii) Suppose that either the assumption of (i) or of (ii) holds. If P1 is the stabilizer in G of a 1-space of the natural module W ކn of G and V is an ކG-module D q affording ˆ, then CV .Q1/ 0 for Q1 Op.P1/. ¤ WD Proof. (i) Suppose that n 5 and ˆ has property .W/. Then we can choose x; y; z from a standard subgroup H GLn 1.q/ or SLn 1.q/ of G. By Lemma 2.4, ˆ H D j has property .Z/, and so does ˆ. (ii) Suppose now that n 4 and ˆ has property .P /. Then we can choose x; y; z k from a standard subgroup H GL4.q/ or SL4.q/ of G. First we consider the case D 0;0 n 4, so that k 2, and let Irr.H/; in particular, .1/ .q4 q/=.q 1/. D D WD 4;q 2 D By Lemma 2.4, has property .Z/. On the other hand, in the notation of the proof of Corollary 2.3, it follows from the arguments in that proof that X Q .2q 1/ 1Q: j D C C O1 2 Hence, choosing x; y Q, we see that 2 (2-2) ˛Œ1 Œ1 0 D D P O for ˛ 1Q and O1 . Also note that P always acts transitively on 1, WD WD 2 no matter if H GL4.q/ or SL4.q/. It follows that ' Q a˛ b for some D j D C integers a; b 0, and so (2-2) implies that 'Œ1 0. D Now we consider the case n 5. If G GLn.q/, then by Proposition 2.1(i), ˆ H D j has property .P2/ and so 'Œ1 0 by the case n 4. Suppose now that G SLn.q/ D D D and set M GLn.q/ G. We can choose a standard subgroup L GL4.q/ in M D Š M such that L G H SL4.q/. By Proposition 2.5(ii) applied to .M; G/, ˆ \ D Š M has property .Pk /, and so .ˆ / L has property .P2/ by Proposition 2.1(i). But M j then .ˆ / H has property .P2/ by Proposition 2.5(i). As we can find a cyclic j subgroup C Cq 1 of L such that L H Ì C and M G Ì C , we see that M Š L D D .ˆ / H ..ˆ H / / H . We can now conclude that ˆH has property .P2/ and so j D j j 'Œ1 0 again by the case n 4. D D (iii) By the results of (i) and (ii), we may assume that 'Œ1 0 for the Brauer D character ' of ˆ. Assume the contrary that CV .Q1/ 0. Note that P1 acts D WEIL REPRESENTATIONS OF FINITE GENERAL LINEAR GROUPS 489 transitively on IBr.Q1/ 1Q . It follows that we can write n f 1 g X (2-3) 'Q c 1 D 1Q IBr.Q1/ 1 ¤ 2 for some integer c > 0. In particular, taking x Q1 we get 2 '.1/ c.qn 1 1/; '.x/ c; D D and so the relation 'Œ1 0 implies that D (2-4) '.y/ c.qn 2 1/: D C On the other hand, as n 4, we can choose an SLn.q/-conjugate y1 P1 of y that 2 projects onto a transvection in GLn 1.q/ under the embedding P1=Q1 , GL1.q/ GLn 1.q/: ! n 2 Such an element y1 acts on IBr.Q1/ 1Q1 with exactly q 1 fixed points. Coupled with (2-3), this implies that n f g n 2 '.y/ '.y1/ .q 1/c; j j D j j Ä contradicting (2-4). 3. The general linear groups

For G GLn.q/ and V an irreducible ކG-module, we will use James’ parametriza- D tion[1986] (3-1) V D.s1; 1/ D.s2; 2/ D.st ; t / G D ı ı ı " for V as given in[Guralnick and Tiep 1999, Proposition 2.4]. Here, si ކ has Pt 2 degree di over ކq and is `-regular, i ki, and n i 1 kidi. Moreover, for any ` D D i j , si and sj do not have the same minimal polynomial over ކq. Each D.si; i/ is ¤ an irreducible module for GLki di .q/, and if t > 1 then V is Harish-Chandra-induced from the Levi subgroup L GL .q/ GL .q/ GL .q/ D k1d1 k2d2 kt st of a certain parabolic subgroup P QL of G. Namely, consider D D.s1; 1/ D.s2; 2/ D.st ; t / ˝ ˝ ˝ as an irreducible L-module, inﬂate it to an irreducible ކP-module U , and then induce to G to get V : V U G. Note that the Harish-Chandra induction is D commutative (with respect to the factors D.si; i/) and transitive. Also, note that the Weil modules of G are precisely D.a;.n 1; 1// and D.a;.n 1// D.b;.1// G, ı " with a; b ކ being `-regular and a b. For the irreducible complex G-modules, 2 q ¤ we will instead use the notation S.s1; 1/ S.s2; 2/ S.st ; t / G: ı ı ı " 490 PHAM HUU TIEP

Proposition 3.1. Theorem B holds for G GL4.q/. D Proof. The implication “(i) (ii)” follows from Corollary 2.3. For the other ) implication, let ˆ be an irreducible ކG-representation possessing property .P2/ and let ' denote its Brauer character. By Corollary 2.6(ii)–(iii), we have 'Œ1 0 D and CV .Q1/ 0, if V is an ކG-module affording ˆ. ¤ (i) First we consider the case ` 0, or more generally, ' lifts to a complex character D of G. Using the character table of G given, e.g., in[Geck et al. 1996], one can check that the relation 'Œ1 0 implies that ' is a Weil character. (Note that the D character table of G was ﬁrst determined in[Steinberg 1951].) ˇ (ii) From now on we may assume that p ` ˇ G and that ' does not lift to a ¤ j j complex character. We use the label for V given in (3-1). Since any irreducible ކX -module of X GL1.q/; GL2.q/ lifts to a complex module, we see that 2 f g D.si; i/ lifts to a complex GL .q/-module if kidi 2. The same also happens ki di Ä if i .ki/; see[Guralnick and Tiep 1999, Corollary 2.6]. Hence, the condition D that V does not lift to a complex module implies that one of the following cases must occur for V as labeled in (3-1): (a) t 1, V D.s; /. D D (b) t 2, V D.a; / D.b;.1// G, deg.a/ deg.b/ 1, 3, and a b. D D ı " D D ` ¤ Let e be the smallest positive integer such that ` .1 q q2 qe 1/. j C C C C (iii) Suppose we are in case (a). Then, by[Kleshchev and Tiep 2010, Theorem 5.4], CV .Q1/ 0 whenever deg.s/ > 1. Hence we must have that deg.s/ 1. By D D [Guralnick and Tiep 1999, Lemma 2.9] without loss we may assume that s 1, D i.e., ˆ is a unipotent representation. Note that ˆ has degree 1 if .4/ and is a D Weil representation if .3; 1/. Now we let , 1 j 5, denote the complex D j Ä Ä unipotent character of G labeled by .4/, .3; 1/, .2; 2/, .2; 12/, and .14/, respectively. Similarly, we let 'j , 1 j 5, denote the Brauer unipotent character of G labeled Ä Ä by .4/, .3; 1/, .2; 2/, .2; 12/, and .14/, respectively. Then 4 5 6 (3-2) 1Œ1 2Œ1 '1Œ1 '2Œ1 0; 3Œ1 q ; 4Œ1 q ; 5Œ1 q : D D D D D D D Recall that we use the notation ı to denote the restriction of a complex character to the set of ` -elements of G, and that 'Œ1 0. 0 D By the results of[James 1990], there are integers x1; x2 such that

'3 x1'1 x2'2: 3ı D C C 4 It follows by (3-2) that '3Œ1 3Œ1 q ; in particular, .2; 2/. D D ¤ Next, by[Guralnick and Tiep 1999, Proposition 3.1], there are nonnegative integers y1; y2; y3 such that y3 1 and Ä '4 y1'1 y2'2 y3'3: 4ı D C C C WEIL REPRESENTATIONS OF FINITE GENERAL LINEAR GROUPS 491

It follows by (3-2) that

4 '4Œ1 4Œ1 y3'3Œ1 q .q y3/ > 0 D D I in particular, .2; 12/. ¤ 4 Finally, let .1 /, i.e., ' '5. Then is 2-divisible and 4-divisible in the D D sense of[Kleshchev and Tiep 2010, Deﬁnition 4.3]. Since CV .Q1/ 0, it follows ¤ by [loc. cit., Theorem 5.4] that e 2; 4. If e 3, then by[James 1990], we have ¤ D ' '3; 5ı D C 4 and so 5Œ1 '3Œ1 q , contrary to (3-2). Thus we must have e 5 > n, and D D so '5 by[James 1990, Theorem 6.4], whence 5Œ1 'Œ1 0, again a D 5ı D D contradiction. (iv) Suppose we are in case (b). By[Guralnick and Tiep 1999, Lemma 2.9], we may assume that a 1. Let 1, 2, and 3 denote the (ordinary) characters of the D irreducible ރG-modules S.1; / S.b;.1// G, with .3/, .2; 1/, and .13/, ı " D respectively. Similarly, let 1, 2, and 3 denote the Brauer characters of the irreducible ކG-modules D.1; / D.b;.1// G, with .3/, .2; 1/, and .13/, ı " D respectively. Using[Geck et al. 1996], we can compute

4 5 (3-3) 1Œ1 0; 2Œ1 q .q 1/; 3Œ1 q .q 1/: D D C D C Note that 1 is a Weil character, and so 1Œ1 0. Using the decomposition D matrix for GL3.q/ [James 1990], we get a nonnegative integer x such that

2 x 1: 2ı D C 4 It follows by (3-3) that 2Œ1 2Œ1 q .q 1/. Furthermore, there are integers D D C 0 y; z 1 such that Ä Ä 3 y 1 z 2: 3ı D C C It follows by (3-3) that

4 2 3Œ1 3Œ1 z 2Œ1 q .q 1/: D We have therefore shown that ' 1, a Weil character. D Proposition 3.2. Let G GLn.q/ with n 5 and let V be an irreducible ކG-module. D Suppose that V has property .W/. Then in the label (3-1) for V , deg.si/ 1 for D all i. Furthermore, either V is a Weil module, or t 1. D Proof. (i) First we consider the case t 1. By Corollary 2.6, CV .Q1/ 0, whence D ¤ deg.s1/ 1 by[Kleshchev and Tiep 2010, Theorem 5.4]. D (ii) From now on we may assume that t 2, so that V is Harish-Chandra- induced. Next we show that deg.si/ 1 for all i. Assume for instance that D 492 PHAM HUU TIEP d1 deg.s1/ > 1. Then we consider a standard subgroup H GLm.q/ of G D D with 2 m n kt dt n 1. Property .W/ implies that all composition factors Ä WD Ä of V H are Weil modules or of dimension 1. We can ﬁnd a parabolic subgroup j P QL with Levi subgroup L H GL .q/. Then note that V can be obtained D D kt dt by inﬂating the irreducible ކL-module A D.st ; t / to P, where ˝ A D.s1; 1/ D.s2; 2/ D.st 1; t 1/ H; WD ı ı ı " and then induce to G. In particular, A is a simple submodule of V H , and clearly A j is not a Weil module as deg.s1/ > 1. Suppose that dim.A/ 1. Since deg.s1/ > 1, D we must have that .m; q/ .2; 2/, deg.s1/ 2, 1 .1/, and t 2. Applying D D D D property .W/ to a standard subgroup GL3.2/ containing H GL2.2/ (as its D standard subgroup) and then restricting further down to H , we see, however, that A deg.s1;.1// cannot occur in V H , a contradiction. D j (iii) Here we show that one of the following holds:

(a) t 3, i .ki/ for all i, and k1; k2; k3 n 2; 1; 1 . D D f g D f g (b) t 2 and i .ki/; .ki 1; 1/ for all i. D 2 f g The arguments in (ii) show that A is a simple submodule of V H . Again by j property .W/, A is either a Weil module or of dimension 1. First we consider the case t 3. Then note that dim.A/ is at least the index of a proper parabolic subgroup of H and so can never be equal to 1. Thus A is a Weil module. The identiﬁcation of Weil modules among the ones labeled in (3-1) now implies that in fact t 3, D i .ki/ for all i 1; 2, and 1 k1; k2 . Interchanging k3 with k1 or k2 in the D D 2 f g above construction and noting that n 5, we then get k3 1 and 3 .1/ as well. D D Suppose now that t 2. The claim is obvious if ki 2, so we consider the D Ä case where k1 3, say. Then A D.s1; 1/ is a Weil module of GL .q/ or of D k1 dimension 1. It follows that 1 .k1/; .k1 1; 1/ , as stated. 2 f g (iv) Abusing the notation, now we use H to denote the standard subgroup H StabG e1; e2;:::; en 1 ކq ; en GLn 1.q/; D h i Š and P Q Ì L to denote the parabolic subgroup D P StabG e1;:::; e ކ ; D h k i q where k k1 and WD L StabG e1;:::; ek ކq ; ek 1;:::; en ކq : D h i h C i Then we can obtain V by inﬂating the irreducible ކL-module B, where B D.s1; 1/ D.s2; 2/ D.st ; t / GLn k .q/; WD ˝ ı ı " WEIL REPRESENTATIONS OF FINITE GENERAL LINEAR GROUPS 493 to an irreducible ކP-module U and then induce to G. Note that PH QH Ì LH D is also a parabolic subgroup of H , where

PH P H; QH Q H; LH L H: WD \ WD \ WD \ H Mackey’s formula implies that V H contains a subquotient V .U / , where j 0 WD 0 U U P . But note that QH Q acts trivially on U , so in fact V is Harish- 0 WD j H Ä 0 Chandra-induced from the ކLH -module B , where B B L . 0 0 WD j H (v) Now we can complete the case t 3. As shown in (iii), in this case we have that D i .ki/ for all i and we may furthermore assume that .k1; k2; k3/ .1; 1; n 2/. D D Repeating the argument in (iv) and using the notation therein, we see that B0 contains a simple subquotient isomorphic to D.s1;.1// D.s2;.1// D.s3;.n 3// GLn 2.q/: ˝ ı " It follows that V H contains a subquotient isomorphic to j D.s1;.1// D.s2;.1// D.s3;.n 3// H; ı ı " which is irreducible, but not a Weil module nor of dimension 1. This contradiction shows that the case t 3 is impossible. D (vi) Finally, we consider the case t 2. As shown in (iii), i .ki/; .ki 1; 1/ ; D 2 f g also, recall that n k1 k2 5. Suppose ﬁrst that k2 k1 2. Then, in the D C notation of (iv), we see that B D.s1; 1/ D.s2; 2/ and so B contains a simple D ˝ 0 subquotient isomorphic to D.s1; 1/ D.s2; / with .k2 1/; .k2 2; 1/ . ˝ 2 f g It follows that V H contains a subquotient isomorphic to j D.s1; 1/ D.s2; / H; ı " which is irreducible, but not a Weil module nor of dimension 1, a contradiction. Hence we may assume that .k1; k2/ .1; n 1/. If 2 .n 1/, then V is a D D Weil module. Assume that 2 .n 2; 1/. Then, in the notation of (iv), we see that D B D.s1; 1/ D.s2;.n 2; 1// and so B contains a simple subquotient isomor- D ˝ 0 phic to D.s1; 1/ D.s2;.n 3; 1//. It follows that V H contains a subquotient ˝ j isomorphic to D.s1;.1// D.s2;.n 3; 1// H; ı " which is irreducible, but not a Weil module nor of dimension 1, again contradicting property .W/.

Proposition 3.3. Let G GLn.q/ with n 5 and let V be an irreducible ކG-module. D Suppose that V has property .W/. Then either V is a Weil module, or dim V 1. D Proof. (i) Consider the label (3-1) for V . By Proposition 3.2 and[Guralnick and Tiep 1999, Lemma 2.9], we may assume that V D.1; /, a unipotent representation; D furthermore, CV .Q1/ 0 by Corollary 2.6. Now V is a subquotient of the ¤ 494 PHAM HUU TIEP reduction modulo ` of the unipotent ރG-module S.1; /. We can ﬁnd a standard subgroup H GLn 1.q/ of G as a direct factor of the Levi subgroup L1 D D H GL1.q/ of P1 Q1L1. By the Howlett–Lehrer comparison theorem[1983, D Theorem 5.9], the Harish-Chandra restriction RG of unipotent characters of G L1 can be computed inside the Weyl group Sn of G, and it is similar for the Harish- Chandra induction RG . For brevity, we denote the character of S.1; / by and L1 the Brauer character of D.1; / by ', with similar notation for other partitions. In this notation, RG . / is the sum of unipotent characters of L labeled L1 1 by .n 1/, where the Young diagram Y ./ of is obtained from the Young ` diagram Y ./ of by removing one removable node. For instance, RG . .n// .n 1/; L1 D G .n 1;1/ .n 1/ .n 2;1/ RL . / ; (3-4) 1 D C RG . .n 2;2// .n 3;2/ .n 2;1/; L1 D C 2 2 RG . .n 2;1 // .n 3;1 / .n 2;1/: L1 D C It is similar for the Harish-Chandra induction; in particular, RG . .n 1// .n/ .n 1;1/; L1 D C (3-5) 2 RG . .n 2;1// .n 1;1/ .n 2;2/ .n 2;1 /: L1 D C C Let be the Brauer character of a simple submodule of the L1-module CV .Q1/. Then is an irreducible constituent of RG .'/, the Brauer L -character of L1 1 CV .Q1/. The above arguments show that is an irreducible constituent of . /ı for some .n 1/, whence ' for some .n 1/. On the other hand, ` D ` property .W/ implies that is a Weil character (while restricted to H ), or has degree 1. As n 5, it follows that .n 1/ or .n 2; 1/. By Frobenius’ D reciprocity, ' is an irreducible constituent of RG .'/, and so of .RG . // L1 L1 ı as well. Restricting (3-5) to `0-elements, we see by[Guralnick and Tiep 1999, Proposition 3.1] that is .n/, .n 1; 1/, .n 2; 2/, or .n 2; 12/. The ﬁrst possibility leads to the principal character, and the second one yields a Weil character. (ii) Here we consider the case .n 2; 2/. Then applying [loc. cit., Proposition 3.1] D to G, we can write .n/ .n 1;1/ . / ' x1. / x2. / ı D C ı C ı for some integers x1, x2. It follows by (3-4) that G .n 3;2/ .n 2;1/ .n 1/ R .' / .1 x2/ .x1 x2/ ı: L1 D C C Applying [loc. cit., Proposition 3.1] to H , we then get RG .'/ '.n 3;2/ x '.n 1/ x '.n 2;1/ L1 D C 10 C 20 WEIL REPRESENTATIONS OF FINITE GENERAL LINEAR GROUPS 495 for some integers x10 , x20 . By the linear independence of irreducible Brauer charac- .n 3;2/ ters, ' is an irreducible constituent of .' / H , contradicting .W/. j (iii) Finally, assume that .n 2; 12/. Then applying [loc. cit., Proposition 3.1] D to G, we have .n/ .n 1;1/ .n 2;2/ . / ' y1. / y2. / y3. / ı D C ı C ı C ı for some integers y1, y2, y3. It follows by (3-4) that G .n 3;12/ .n 3;2/ .n 2;1/ .n 1/ R .' / y3 .1 y2 y3/ .y1 y2/ ı: L1 D C C Applying [loc. cit., Proposition 3.1] to H , we then get

2 RG .'/ '.n 3;1 / y '.n 1/ y '.n 2;1/ y '.n 2;2/ L1 D C 10 C 20 C 30 for some integers y10 , y20 , y30 . By linear independence of irreducible Brauer charac- .n 3;12/ ters, ' is an irreducible constituent of .' / H , again contradicting .W/. j We note (without giving proof, since we do not need it subsequently) that property .Z/ can also be used to characterize Weil representations of GLn.q/ with n 5. Proposition 3.4. Theorem A holds for G GLn.q/ with n 5. D Proof. The implication “(i) (ii)” follows from Corollary 2.3. In fact, by applying ) Corollary 2.3 successively, we see that (i) also implies (iii) and (iv). Also, note that (iv) obviously implies (iii), and (ii) implies (i) by Proposition 3.3. It remains to show that (iii) implies (i). We proceed by induction on n 5, with the induction base n 5 (so that m 4) already established in Proposition 3.3. For D D the induction step n 6, consider a chain of standard subgroups H GLm.q/ L GLn 1.q/ < G GLn.q/: D Ä D D Let ‰ be any composition factor of ˆ L. According to (iii), every composition j factor of ‰ H is either a Weil representation or has dimension 1. Hence, by the j induction hypothesis applied to L, we see that either ‰ is a Weil representation or deg ‰ 1. Thus ˆ has property .W/, and so we are done by Proposition 3.3. D Proposition 3.5. Theorem B holds for G GLn.q/ with n 4. D Proof. The implication “(i) (ii)” follows from Corollary 2.3. For the other impli- ) cation, let G GLn.q/ with n 4 and let ˆ be an irreducible ކG-representation D with property .P / for some 2 k n 2. By Proposition 3.1, we may assume that k Ä Ä n 5 and consider a standard subgroup H GL4.q/ of G. By Corollary 2.2(i), Š every composition factor ‰ of ˆ H has property .P2/. By Proposition 3.1, either ‰ j is a Weil representation or deg ‰ 1. Thus ˆ fulﬁlls condition (iii) of Theorem A D for G and with m 4. Hence ˆ is either a Weil representation or has degree 1 by D Proposition 3.4. 496 PHAM HUU TIEP

4. The special linear groups

Proof of Theorems A and B. By Propositions 3.4 and 3.5, it sufﬁces to prove the theorems for S SLn.q/. Also, by Corollary 2.3, it sufﬁces to prove the implication D “(ii) (i)” (as the implication “(iii) (i)” of Theorem A can then be proved using ) ) the same arguments as in the proof of Proposition 3.4). Let P .P /; .W/ and 2 f k g let U be an irreducible ކS-module with property P. We consider S as the derived G subgroup of G GLn.q/. By Proposition 2.5(ii), a simple submodule V of U has D property P. Applying Proposition 3.5 if P .P /, respectively Proposition 3.4 if D k P .W/, to V , we see that V is a Weil module or dim V 1. As U is an irreducible D D constituent of V S , we conclude that U is a Weil module or has dimension 1. j We note that one could try to prove the complex case of Theorem A for GLn.q/ using the results of[Zelevinsky 1981] or[Thoma 1971]. We conclude by the following example showing that Weil representations of GLn.q/ and SLn.q/ do not admit a “middle-free” characterization in the spirit of[Guralnick et al. 2002, Corollary 12.4].

Example 4.1. Let G GLn.q/ or SLn.q/ with n 3. Consider the natural module D W e1;:::; en ކ and let D h i q P StabG e1 ކq ; e1;:::; en 1 ކq : D h i h i Note that P NG.Z/ for a long-root subgroup Z of G. Also, we may assume D that (a G-conjugate of) the long-root element x deﬁned in Section 2 is contained in Z. Suppose that an irreducible ކG-module V has no middle with respect to Q Op.P/ as in[Guralnick et al. 2002, Corollary 12.4], i.e., D CV .Z/ CV .Q/: D It follows that V CV .Q/ ŒV; Z: D ˚ Let ' denote the Brauer character of V . Then we have '.x/ a qn 2b; D n 2 where dim CV .Q/ a and dimŒV; Z q .q 1/b. But note that Q Z contains D D n a G-conjugate x0 of x, and '.x / a: 0 D It follows that b 0, V CV .Q/, Q acts trivially on V , and so dim V 1. D D D Acknowledgements

The author is grateful to Alexandre E. Zalesski for bringing the question of rec- ognizing Weil representations of ﬁnite general and special linear groups to his attention. Part of the work was done while the author was visiting the Institute for WEIL REPRESENTATIONS OF FINITE GENERAL LINEAR GROUPS 497

Advanced Study, Princeton. It is a pleasure to thank Peter Sarnak and the Institute for generous hospitality and a stimulating environment. The author is also grateful to the referee for careful reading and helpful comments on the paper.

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Received March 9, 2015. Revised July 22, 2015.

PHAM HUU TIEP DEPARTMENT OF MATHEMATICS UNIVERSITYOF ARIZONA TUCSON, AZ 85721 UNITED STATES [email protected] PACIFIC JOURNAL OF MATHEMATICS msp.org/pjm

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Volume 279 No. 1-2 December 2015

In memoriam: Robert Steinberg

Robert Steinberg (1922–2014): In memoriam 1 V. S. VARADARAJAN Cellularity of certain quantum endomorphism algebras 11 HENNING H.ANDERSEN,GUSTAV I.LEHRER and RUIBIN ZHANG Lower bounds for essential dimensions in characteristic 2 via orthogonal representations 37 ANTONIO BABIC and VLADIMIR CHERNOUSOV Cocharacter-closure and spherical buildings 65 MICHAEL BATE,SEBASTIAN HERPEL,BENJAMIN MARTIN and GERHARD RÖHRLE Embedding functor for classical groups and Brauer–Manin obstruction 87 EVA BAYER-FLUCKIGER,TING-YU LEE and RAMAN PARIMALA

On maximal tori of algebraic groups of type G2 101 CONSTANTIN BELI,PHILIPPE GILLE and TING-YU LEE On extensions of algebraic groups with finite quotient 135 MICHEL BRION Essential dimension and error-correcting codes 155 SHANE CERNELE and ZINOVY REICHSTEIN Notes on the structure constants of Hecke algebras of induced representations of finite Chevalley groups 181 CHARLES W. CURTIS 2015 Complements on disconnected reductive groups 203 FRANÇOIS DIGNE and JEAN MICHEL Extending Hecke endomorphism algebras 229 JIE DU,BRIAN J.PARSHALL and LEONARD L.SCOTT Products of partial normal subgroups 255 ELLEN HENKE 1-2 No. 279, Vol. Lusztig induction and `-blocks of finite reductive groups 269 RADHA KESSAR and GUNTER MALLE Free resolutions of some Schubert singularities 299 MANOJ KUMMINI,VENKATRAMANI LAKSHMIBAI,PRAMATHANATH SASTRY and C. S. SESHADRI Free resolutions of some Schubert singularities in the Lagrangian Grassmannian 329 VENKATRAMANI LAKSHMIBAI and REUVEN HODGES Distinguished unipotent elements and multiplicity-free subgroups of simple algebraic groups 357 MARTIN W. LIEBECK,GARY M.SEITZ and DONNA M.TESTERMAN Action of longest element on a Hecke algebra cell module 383 GEORGE LUSZTIG Generic stabilisers for actions of reductive groups 397 BENJAMIN MARTIN On the equations defining affine algebraic groups 423 VLADIMIR L.POPOV Smooth representations and Hecke modules in characteristic p 447 PETER SCHNEIDER On CRDAHA and finite general linear and unitary groups 465 BHAMA SRINIVASAN Weil representations of finite general linear groups and finite special linear groups 481 PHAM HUU TIEP The pro-p Iwahori Hecke algebra of a reductive p-adic group, V (parabolic induction) 499 MARIE-FRANCE VIGNÉRAS Acknowledgement 531

0030-8730(2015)279:1;1-1