January 30, 207 Dstruton det x s on p-adc matrces aul arrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ Let e a p-adc feld wth ntegers o, local parameter, and resdue feld cardnalty q. Let A = M n e the -vectorspace of n-y-n matrces over k, and = L n. Let u s e the tempered dstruton u s f = det x s fx d x Schwartz functon f on A, for Res where d x denotes a Haar measure on. Up to constants, for addtve Haar measure d + x on A, d x = d + x/ det x n. or revty, wrte x for det x when possle. [0.] Convergence The ntegral defnng u s converges asolutely n Res > n : Recall the Iwasawa decomposton = wth the paraolc sugroup of upper-trangular matrces. Snce s open n, Haar measure on restrcted to s Haar measure on. Recall the ntegral formula fg dg = fpk dp dk up to normalzaton, wth left Haar measure on In Lev-Malcev coordnates N M =, wth N the unpotent radcal and M dagonal matrces, up to normalzaton, left Haar measure on s x 2... x n y y 2 d...... xn,n yn = dx 2 dx 3... dx n,n dy y +n dy 2 y 2 +n 3 dy 3 y n +n 5... dy n y n n wth addtve Haar measures n the coordnates. or χ the characterstc functon of M n o, and Res, up to normalzaton constants, u s χ = χg det x s dg = χp det p s dp = y j s dy j dx j j o k <j y j o y j +n 2j = y j s j dy j = y j o k y j +n 2j j s n j dy j = j o k y j q s n j j Thus, u s χ converges asolutely n Res > n, admts a meromorphc contnuaton, and defntely lows up as s n +. Usng the homogenety of u s, the ntegral expreson for u s f for any Schwartz functon f s domnated y the ntegral for u s χ, so u s gves a tempered dstruton n Res > n. [0.2] Meromorphc contnuaton and resdues of u s χ The outcome of the computaton of u s χ to understand convergence also gves a meromorphc contnuaton n s, wth smple poles and non-zero resdues at s =, 0 and at ponts dfferng y nteger multples of 2π/ log q from these. [0.3] Meromorphc contnuaton of u s f or an artrary Schwartz functon f the value u s f can e meromorphcally contnued smlarly, as follows. rst, snce u s s rght -nvarant, frst average f on the rght over, and then use a Lev-Malcev decomposton: det x s fx d x = det p s fxk dk d left p = m s n... m n s n f nm dn dm y n
aul arrett: Dstruton det x s on p-adc matrces January 30, 207 where f s the averaged f. Snce f s tself a Schwartz functon, t s a fnte lnear comnaton of monomals ϕx = ϕ j x j j of Schwartz functons ϕ j n the coordnates x j. Of course, the support of ϕ j must nclude 0 for > j, or else ϕp = 0. or < j, the relevant ntegral s ϕ j x j m j dx j = m j ϕ j x j dx j The whole s ϕ j 0 ϕ j x j dx j ϕ m m s n 2 d m >j <j = ϕ j 0 ϕ j x j dx j ϕ t t s n d t >j <j The frst two products are constants. Each ntegral n the last product s an Iwasawa-Tate local zeta ntegral: when the support of ϕ does not nclude 0, t s a polynomal n q s, and when the support of ϕ does nclude 0, the zeta ntegral s a sum of a polynomal n q s and a constant multple of q s j. nte sums of such expressons admt meromorphc contnuatons wth poles at most at s = n, n 2,..., 2,, 0 and ponts dfferng from these y nteger multples of 2π/ log q. oles, f any, are smple. Thus, v s f = q s n... q s u s f has a meromorphc for every Schwartz functon f. That s, v s s weakly holomorphc. Weak holomorphy mples strong holomorphy for vector-valued functons wth values n a quas-complete locally convex topologcal vector space. Tempered dstrutons are such. Thus, v s s a holomorphc tempered-dstruton-valued functon of s C. In partcular, the resdues of u s at poles are tempered dstrutons. [0.4] Support of resdues or f a Schwartz functon wth support nsde, the meromorphc scalarvalued functon u s f s entre. Thus, the resdues of u s at s = n, n 2,..., 0 are tempered dstrutons supported on the set A <n of matrces of less-than-full rank. [0.5] Unqueness and exstence of equvarant dstrutons The standard argument shows that, gven s C, there s a unque tempered dstruton on such that uaxb = det A det B s ux, snce acts transtvely on tself. The tempered dstruton s gven y the ntegral for u s n the range of convergence, and y meromorphc contnuaton otherwse. Let = {g : det g = } The product acts transtvely on the set A r of matrces of a gven rank r < n y g hx = g xh. The sotropy group of r 0 E r = 0 0 n r s A a 0 H r = { : D L 0 D c D n r o, a, A L r, det a = det A = det D } Both H r and are unmodular, so there s a unque -nvarant measure on A r /H r, and ntegraton aganst ths measure gves the unque -nvarant dstruton on Schwartz functons supported on the set A r of matrces of rank r. 2
aul arrett: Dstruton det x s on p-adc matrces January 30, 207 At the same tme, s already transtve on A r, so up to scalars there s unque -nvarant measure and correspondng dstruton. We can easly wrte a formula for t n terms of Eucldean coordnates, namely u r f = r r x 0 f x n r r 2 0 n r dx 2 dx dk By the unqueness of -nvarant functonal, ths ntegral formula must also e a -nvarant functonal. Smlarly, the -nvarant form of that ntegral must gve the same functonal: u r f = r r x x fk 2 0 0 r n r n r k dx 2 dx dk Equalty up to constants follows from unqueness, and the constant s ecause the two ntegrals agree on the characterstc functon of Λ. These ntegrals converge asolutely, so extend to tempered dstrutons on the whole Schwartz space. Changng varales n the frst ntegral expresson, the equvarance under the full group = L n s u r x fax Changng varales n the second ntegral expresson, u r x fxb = det A r u r f for A L n = det B r u r f for B L n The resdue of u s at s = r < n s supported on the set A r of matrces of rank r, and has the same equvarance under as does u r. Suggestng that, up to a constant, the dstruton u r s the resdue of u s at s = r < n. Indeed, on Schwartz functons supported on A r, the unqueness result just aove does show that the resdue of u s at r s a constant multple of u r. The ntegral expresson for u r specfes t on the whole Schwartz space. The appearance of the resdue as a resdue specfes t on the whole Schwartz space. The dfference v of sutale multples vanshes on Schwartz functons supported on A r. Ths dfference restrcted to Schwartz functons supported on A r s -nvarant, so must e a multple of u r. However, the -equvarance does not match that of u r, so ths restrcton to A r s 0. Smlarly, the restrcton of v to Schwartz functons supported on A r 2 must e a multple of u r, and the equvarance forces t to e 0. Contnung, we fnd that the resdue of u s at s = r < n s a multple of u r. [0.6] Non-extendalty of det x for L 2 In the small example = L 2, a relatvely elementary argument shows that u f = x fx d x has no extenson from Schwartz functons supported on to the whole space of Schwartz functons on A. Specfcally, we clam that any tempered dstruton u wth the homogenety property ur g f = det g uf wth R g fx = fxg s supported on A. Ths wll follow from the Hecke operator dentty proven elow ch = ch Λ + q ch Λ T p ch Λ where ch X s the characterstc functon of a set, Λ = M 2 o, and T p s the Hecke operator ncarnated as T p fx = ch D g fxg dg wth D n = {: g M 2 o, det g = q n } 3
aul arrett: Dstruton det x s on p-adc matrces January 30, 207 Indeed, a tempered dstruton u wth the ndcated homogenety property restrcted to Schwartz functons on s c u for some constant c, y unqueness. We show that c = 0. The nteracton of Hecke operator and u s easly determned: ut p f = u ch D g R g f dg = ch D g ur g f dg = ch D g uf det g dg = uf q ch D g dg = uf q q + Applyng u to the dentty of characterstc functons gves uch = uch Λ + q uch Λ ut p ch Λ and then c u ch = uch Λ + q q 2 uch Λ q q + uch Λ = uch Λ + q q 2 q q + = uch Λ 0 yeldng c = 0. To prove the dentty ch = ch Λ + q ch Λ T p ch Λ we explcate the acton of T p on the functons ch Dn, snce ch Λ = n 0 ch Dn + ch A As n the classcal context, T p ch Dn x = ch D g ch Dn xg dg s certanly 0 unless x D n+. Left modulo, D n+ has representatves 0 n+ wth mod n+ Smlarly for g rght modulo, equvalently, for g D left modulo, take representatves g 0 =, wth 0 0 mod and then and xg = g = 0, 0 0 n+, / 0 / 0 n wth mod wth 0 n + ven x, the ntegral produces the value for g such that xg M 2 o. The frst case gves a value exactly for. The second famly gves for n and o, or for = 0 and = mod. In summary, + q for n and o for n and o T p ch Dn 0 n+ = for = 0 for = n + 4 0 n + and mod n+
That s, aul arrett: Dstruton det x s on p-adc matrces January 30, 207 T p ch Dn = { q chdn + ch D n+ for n ch D for n = 0 Ignorng sngular matrces snce the outcome T p ch Λ s guaranteed to e a Schwartz functon, T p ch Λ = T p ch Do + ch Dn n = ch D + n q ch Dn + ch Dn+ = q ch Λ + ch Λ ch snce D o =. Ths rearranges to the asserted dentty, from whch the non-extendalty follows. 5