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Addchar53-1 the ز¥Theorem in This Section, We Will addchar53-1 The Ò¥ theorem In this section, we will consider in detail the following general class of "standard inputs" [rref defn of std input]. We work over a finite field k of characteristic p, in which the prime … is invertible. We take m=1, ≠ a nontrivial ä$… -valued additive character ¥ of k, 1 K=Ò¥(1/2)[1] on ! , an integern≥1, V=!n, h:V¨!l the functionh=0, LonVaperverse, geometrically irreducible sheaf which is “- pure of weight zero, which in a Zariski open neighborhood U0 of the n origin in ! is of the form Ò[n] forÒanonzero lisse ä$…-sheaf on U0, an integere≥3, (Ï, †) = (∏e, evaluation), for ∏e the space of all k-polynomial functions on !n of degree ≤ e. Statements of the Ò¥ theorem Theorem Take standard input of the above type. Then we have the following results concerningM=Twist(L, K, Ï, h). 1) The object M(dimÏ0/2) onÏ=∏e is perverse, geometrically irreducible and geometrically nonconstant, and “-pure of weight zero. 2) The Frobenius-Schur indicator of M(dimÏ0/2) is given by: geom FSI (∏e, M(dimÏ0/2)) =0,ifpisodd, = ((-1)1+dimÏ0)≠FSIgeom(!n, L), if p= 2. 3) The restriction of M(dimÏ0/2) to some dense open set U of ∏e is of the form ˜(dimÏ/2)[dimÏ] for ˜ a lisse sheaf on U of rank N := rank(˜|U) n ≥ (e-1) rank(Ò|U0), if e is prime to p, n n n ≥ Max((e-2) , (1/e)((e-1) + (-1) (e-1)))rank(Ò|U0), if p|e. addchar53-2 4) The Frobenius-Schur indicator of ˜|U is given by FSIgeom(U, ˜|U) =0,ifpisodd, geom n n geom = FSI (! ,L)=(-1) FSI (U0,Ò),ifp=2. 5) Suppose in addition that one of the following five conditions holds. a)p≥7, b)n≥3, c)p=5ande≥4, d)p=3ande≥7, e) p=2ande≥7. Then we have the following results concerning the group Ggeom,˜|U. geom 5A) If FSI (U, ˜|U) = 0, the group Ggeom,˜|U contains SL(N). geom 5B) If FSI (U, ˜|U) = -1, the group Ggeom,˜|U is Sp(N). geom 5C) If FSI (U, ˜|U) = 1, the group Ggeom,˜|U is either SO(N) or O(N). Proof of the Ò¥ theorem Recall that ∏e is d-separating for d=e+1≥4[rref early in newtry], and that * 1 Hc (! ‚käk, K) = 0. Because the lisse, rank one sheaf Ò¥ is of orderp=char(k) as a 1 character of π1(! ‚käk), it is geometrically self dual if and only if p=2, in which case it is orthogonally self dual. So the Frobenius- 1 Schur indicator ofK=Ò¥(1/2)[1] on ! is given by FSIgeom(!1,K)=0,ifp=char(k) is odd, = -1, if if p = char(k) =2. We first prove 1), that M(dimÏ0/2) onÏ=∏e is perverse, geometrically irreducible, and “-pure of weight zero. This depends on the following compatibility, Compatibility Lemma For anye≥1,thek-morphism n £ eval :! ¨(∏e) vÿeval(v), is a closed immersion. In terms of the Fourier transform on the addchar53-3 £ target space (∏e) , b £ b FT¥ :D c((∏e) ,ä$…)¨D c(∏e,ä$…), we have Twist(L, K, Ï, h) = FT¥(eval*(L))(1/2), i.e., Twist(L, K, Ï, h)(dimÏ0/2)=FT¥(eval*(L))(dimÏ/2). n proof Pick coordinate functions x1,...,xn on ! . Then the map eval n sends a point v in ! with coordinates (v1,..., vn)tothe vector of all monomials in the vi of degree at most e. Just looking at the monomials of degree one, the vi themselves, shows that we have a 1 closed immersion. AsK=Ò¥(1/2)[1] on ! , the statement concerning Fourier transform is a tautology. QED We exploit this compatibility as follows. The object L is perverse and geometrically irreducible and “-pure of weight zero on !n, with support all of !n. Since n ≥ 1, L does not have punctual support. Since eval is a closed immersion, the object eval*Lisperverse and £ geometrically irreducible and “-pure of weight zero on (∏e) , and does not have punctual support. Therefore, by the miraculous properties [rref Ka-Lau 2.2.1, 2.3.1.r ef Laumon Constantes for involutivity?] FT¥(eval*(L)) is perverse and geometrically irreducible and “-pure of weight dim∏e = dimÏ on ∏e, and it is not geometrically constant. This proves 1). Statements 2), 3), and 4), except for the rank estimate in 3), result immediately from applying the Higher Moment Theorem [rref newtry, higher moment thm, third version] withd=e+1≥4,cf. the beginning of [rref explain]. In view of Larsen's Alternative [rref explain], the conclusions 5A), 5B) and 5C) hold whenever Ggeom,˜|U is not finite. So in order to prove 5), it suffices to show that Ggeom,˜|U is not finite if any of the conditions 5a) through 5e) holds. We now explain how to prove the rank estimate in part 3), and how to show that Ggeom,˜|U is not finite if any of the conditions 5a) through 5e) holds. The proof is based upon exploiting a homothety addchar53-4 contraction argument and the Semicontinuity Theorem to reduce to the special case when the object L on !n is geometrically constant. We then treat that case separately. Interlude::: the homothety contraction method With an eye to later applications, we will give the homothety contraction method in slightly greater generality than is required for the Ò¥ theorem. We work over an arbitrary field k in which … is invertible. We fix integersm≥1andn≥1,andaperverse sheaf K on !m.We assume that i m Hc (! ‚äk, K[m])=0fori>-n. We take V:=!n, h:V¨!m the function h=0. We take for (Ï, †)=aspace of !m-valued functions on !n which contains the constants and is quasifinitely difference- separating on !n.Weassume further that (Ï, †) is stable by homothety in the following sense. First of all, we are given a #- i grading of the k-space Ï, say Ï = ·i Ï . Using the grading, for every ≠ k-algebra R we define an action of the group R on Ï‚kR: for t in ≠ [i] i R ,andforf=‡f in Ï‚kR=·i Ï ‚kR, we define i [i] ft := ‡i t f in Ï‚kR. What we require is that under the natual homothety action of R≠ on !n(R), for any point v in !n(R), for any point t in R≠, and for anyfinÏ‚kR, we have f(tv) =ft(v). For example, if m=1, the for anye≥1,wecantake for Ï the space n ∏e of all polynomial functions on ! of degree at most e, with the usual grading by degree. And for general m, if we pick m integers ei ≥1,wecan take for Ï the space ° ∏ of m-tuples of polynomials i ei n functions on ! , the i'th being of degree at most ei.Inthis case, an addchar53-5 m-tuple (f1, ..., fm)offunctions is said to be homogeneous of some given degree d if each fi is homogeneous of that degree. [Remember that we perform addition and scalar multiplication of m-tuples of functions componentwise, so that [i] [i] [i] [i] (‡i f1 ,..., ‡i fm )=‡i (f1 ,..., fm )] We fix a perverse sheaf L on V := !n.Inview of the hypothesis on K, namely i m Hc (! ‚äk, K[m])=0fori>-n, i n and the fact that Hc (! ‚äk, L[n]])=0fori>0,weseethat whatever the perverse sheaf L, the objectM=Twist(L,K,Ï,h=0) is perverse on Ï. Now consider the homothety action of !1 on !n, hmt: !1≠!n ¨!n, (t, v) ÿ tv. Lemma For L a perverse sheaf on V := !n, the object L(tv)[1] := hmt*(L)[1] on !1≠!n lies in pD≥0. proof This is an instance of [rref explain,"first fact", page 8]. QED Lemma For L perverse on V := !n, and for K perverse on !m, the object L(tv)‚K(f(v))[1+dimÏ0] on !1≠!n≠Ï lies in pD≥0. m proof Write Ï as Ï0≠! , with coordinates (f0, a). In coordinates (t, v, f0,a)on 1 n 1 n m ! ≠! ≠Ï=! ≠! ≠Ï0≠! , L(tv)‚K(f(v))[1+dimÏ0]is L(tv)‚K(f0(v) + a)[1+dimÏ0]. 1 n m Under the automorphism ß of ! ≠! ≠Ï0≠! given by (t, v, f0,a)ÿ(t,v, f0,a-f0(v)), the pullback by ß of L(tv)‚K(f(v))[1+dimÏ0]isthe object L(tv)‚K(a)[1+dimÏ0], addchar53-6 which is the external tensor product of L(tv)[1] on !1≠!n, ä$…[dimÏ0]onÏ0, Kon!m. Each external tensoree is in pD≥0, hence also the external tensor product. Since the property of lying in pD≥0 is invariant under pullback by automorphisms, we find that L(tv)‚K(f(v))[1+dimÏ0] lies in pD≥0. QED Consider the morphism 1 n 1 pr1,3 :!≠! ≠Ϩ! ≠Ï, (t, v, f) ÿ (t, f), and the object L(tv)‚K(f(v))[1+dimÏ0] p ≥0 1 n 1 in D on ! ≠! ≠Ï. Define the object Mdef =Mdef(t, f) on ! ≠Ï by Mdef := R(pr1,3)~(L(tv)‚K(f(v))[1+dimÏ0]). [We view Mdef as a deformation of the object M on Ï, whence the name.] 1 p ≥0 Corollary The object Mdef on ! ≠Ï lies in D . proof This is just the fact that for π an affine morphism, Rπ~ preserves pD≥0.
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