By HENRY H. KIM and KYU-HWAN LEE

Transcription

1 SPHERICAL HECKE ALGEBRAS OF SL OVER -DIMENSIONAL LOCAL FIELDS By HENRY H. KIM and KYU-HWAN LEE Absrac. In hs paper, we sudy sphercal Hece algebras of SL over wo dmensonal local felds. In order o defne he convoluon produc, we mae explc use of cose decomposons. We also consder sphercal Hece algebras of he orus of SL and consruc he Saae somorphsm beween wo sphercal Hece algebras. In order o defne he Saae somorphsm, we use he nvaran measure on wo-dmensonal local felds wh values n RX consruced by I. Feseno. Inroducon. The Saae somorphsm or Saae parameers play an mporan role n he Langlands program. Especally, he local L-funcons of sphercal represenaons are defned usng Saae parameers. More precsely, le G be a conneced spl reducve algebrac group defned over a local feld F. Then he Saae somorphsm gves one-o-one correspondence beween he se of equvalence classes of sphercal represenaons of GF and he se of sem-smple conjugacy classes n he L-group ĜC see, for example [Ca]. Le π be a sphercal represenaon of GF wh he sem-smple conjugacy class π n ĜC, and le r : ĜC GL N C be a fne-dmensonal represenaon. Then he local L-funcon s defned by Ls, π, r = de I r π q s 1, where q s he cardnaly of he resdue feld of F. The sudy of sphercal represenaons s exacly he sudy of he sphercal Hece algebras, he se of C-valued compacly-suppored funcons whch are b-nvaran under he acon of he maxmal compac subgroup. On he oher hand, here s anoher mporan algebra, namely, he Iwahor Hece algebra; he one aached o he Iwahor subgroup. I s well nown by Bernsen s heorem ha sphercal Hece algebras are he cener of Iwahor Hece algebras. Ths fac plays an mporan role n he represenaon heory of p-adc groups. Ths paper arose n an aemp o defne local L-funcons for affne Kac- Moody groups over a local feld, whch are aached o exended Dynn dagrams. Manuscrp receved Aprl 8, 004, revsed May 1, 004. Research of he frs auhor suppored by an NSERC gran and an AMS Cenennal Fellowshp; and research of he second auhor suppored n par by KOSEF Gran #R Amercan Journal of Mahemacs ,

2 138 HENRY KIM AND KYU-HWAN LEE Kac-Moody groups are nfne dmensonal groups whch are no locally compac. Hence here are no Haar measures. Affne Kac-Moody groups over a local feld are closely relaed o spl smple algebrac groups over a -dmensonal local feld. The class of such felds ncludes fne exensons of he felds Q p, F p 1 and Q p {{}} for he defnon see [LF]. Kapranov [Ka] suded an analogue of Iwahor Hece algebras for ceran cenral exensons of spl smple algebrac groups over -dmensonal local felds. He used an analyc connuaon echnque and he resdue consrucon of he Cheredn algebra whch does no use generaors and relaons. He proved ha ndeed he algebra of Hece operaors assocaed wh double coses s somorphc o he Cheredn algebra. By analogy wh Bernsen s heorem, we could consder he cener of he Cheredn algebra. The problem s ha he cener conans an nfne sum and so we run no a convergence problem. In hs paper, we sudy sphercal Hece algebras of SL F wh respec o he maxmal compac subgroup SL O, where F s a -dmensonal local feld and O s he rng of negers wh respec o he -dmensonal valuaon. There are several dffcules we need o overcome. Frs, snce he group s no locally compac, we need o fnd an analogous noon of funcons wh compac suppor n he classcal case. We wll consder funcons whch are zero ousde ceran unon of double coses, nroducng he concep of wegh See 3.6. Smlarly, we defne sphercal funcons on he orus T. Second, snce here s no Haar measure, we need o defne he convoluon produc n a combnaoral way. Ths wll be done by usng an explc cose decomposon of a double cose no rgh coses. Because a double cose of K = SL O s an uncounable unon of rgh coses of K, we need o mae he convoluon produc o be zero ousde ceran unon of double coses See 3.9. Thrd, when we wan o defne he Saae somorphsm, mang he classcal consrucon by usng he negral on he unpoen radcal UF, we need measure on UF. We use he nvaran measure consruced by I. Feseno. However, snce s RX-valued, we have o defne he negral o be zero ousde ceran double coses o mae he resulng funcon o be C-valued See 5.5. We show ha he sphercal Hece algebra HG, K s a commuave algebra, generaed by hree elemens. Here generaed means ha we mus allow ceran nfne sums. We explan brefly he conens of hs paper. In Secon 1, we brefly revew he heory of an nvaran RX-valued measure on F consruced by I. Feseno. We noe ha he addve group of a -dmensonal local feld s no locally compac wh respec o s opology and herefore by Wel s heorem, here s no nonrval Haar measure on. In Secon, we generalze he Caran Decomposon o he case of groups over -dmensonal local felds. In Secon 3, we defne he sphercal Hece algebra of SL F wh respec o he subgroup SL O, and s convoluon produc. Snce we have no ye consruced an nvaran measure on SL F, we need o defne he convoluon produc n a combnaoral way. However, we hope o consruc an nvaran measure so ha we may defne

3 SPHERICAL HECKE ALGEBRAS OF SL 1383 he convoluon produc as an negral. In fac, our defnon of he convoluon produc was movaed by such a formula See Remar In Secon 4, we consruc he sphercal Hece algebra of he orus of SL F. The fnal secon s devoed o he consrucon of he Saae somorphsm beween wo sphercal Hece algebras n Secons 3 and 4. In he defnon of hs algebra somorphsm, we use Feseno s measure and negraon heory n Secon 1. There are many quesons o be answered such as deermnng he srucure of sphercal Hece algebras, he sudy of sphercal represenaons of GF, and an analogue of he sem-smple conjugacy classes. We hope o generalze our resuls such as sphercal Hece algebras and he Saae somorphsms o any spl semsmple algebrac groups over -dmensonal local felds n he near fuure. Acnowledgmens We would le o han Professor Feseno for many useful remars and explanaon of hs measure. Thans are due o Dr. J. Gordon and Professor M. Kapranov for her help. Thans are also due o he referee for many commens. 1. Invaran measure on -dmensonal local felds. In hs secon, we brefly revew he heory of he nvaran measure on -dmensonal local felds defned by I. Feseno [Fe]. We refer he reader o [LF] for he defnon and properes of a -dmensonal local feld. Le F =F be a wo dmensonal local feld wh he frs resdue feld F 1 and he las resdue feld F 0 =F q ofq elemens. We denoe by v 1 he dscree valuaon of ran one of F and by v 1 he dscree valuaon of F 1. Also fx a dscree valuaon v : F Z of ran wo. Recall ha Z s endowed wh he lexcographc orderng from he rgh. Le 1, be local parameers wh respec o he valuaon v. Wh respec o he -dmensonal valuaon v, we have he rng of negers O of F, s maxmal deal M and he group of uns U. These do no depend on he choce of v. Smlarly, for he feld F 1, we use he noaons O 1, M 1 and U 1, respecvely. Also, we denoe by O 1 he rng of negers of F wh respec o he dscree valuaon v 1 of ran one, so generaes he maxmal deal M 1 of O 1. There are naural projecons p : O O/M = F 0, p 1 : O 1 O 1 /M 1 = F 1 and p 1 : O 1 O 1 /M 1 = F 0. Example 1.1. Assume ha F = Q p wh local parameers 1 =. Then = p and F 1 = Q p, F 0 = F p, O = Z p + Q p [[]], M = pz p + Q p [[]], U = Z p + Q p [[]], O 1 = Q p [[]], M 1 = Q p [[]], O 1 = Z p, M 1 = pz p, U 1 = Z p.

4 1384 HENRY KIM AND KYU-HWAN LEE In he followng defnon, we specfy a famly of measurable ses. Defnon 1.. A subse of F s called dsngushed f he se s of he form α + 1 j O, α F,, j Z. We denoe by A he mnmal rng conanng all dsngushed subses of F. Alernavely, dsngushed ses are shfs of fraconal prncpal O-deals of F, and A s he mnmal rng whch conans ses α + j p 1 1 S, j Z, where S s a compac open subse n F 1. A sequence { sm X } m N n CX converges o 0fs m 0 for each and here s 0 such ha s m = 0 for all < 0 and for all m. We defne a lnear map res : CX C by a j X j a. A seres c n, c n CX, s called absoluely convergen f converges and res c n converges for every. LEMMA 1.3. [Fe] There s a unque measure µ on F wh values n RX whch s shf nvaran and fnely addve on A such ha µ =0and µ 1 j O=q X j for, j Z. If S s a compac open subse n F 1, and µ 1 s he normalzed Haar measure on F 1 such ha µ 1 O 1 =1,wegeµ j p 1 1 S = Xj µ 1 S. We defne a wo dmensonal module on F by 0 =0, 1 j u = q X j for u U. Then we see ha µαa = α µa for A Aand α F. Le R F be he vecor space generaed by funcons f = c n char An, wh counably many dsjon dsngushed ses A n, and c n C such ha c n µa n converges absoluely, and by funcons g whch are zero ousde fnely many pons. Then we defne he negrals fdµ = c n µa n and gdµ =0. See [Fe] for he well-defnedness. Remar 1.4. The measure µ s counably addve n he followng sense. Assume ha A As a dsjon unon of A A =1,,... Then we have µa = µa, whenever µa s absoluely convergen n RX. For more properes of he measure µ on F, we refer he reader o [Fe], where one can also fnd he defnon and dscusson of local zea negrals on opologcal K - groups of F. A generalzaon of he measure and harmonc analyss o hgher dmensonal local felds has been developed n [Fe1].

5 SPHERICAL HECKE ALGEBRAS OF SL Caran decomposon of G. We prove he Caran Decomposon wh respec o he -dmensonal local feld srucure for a conneced spl sem-smple algebrac group. Le G be a conneced spl sem-smple algebrac group defned over Z. We fx a maxmal orus T and a Borel subgroup B such ha T B G, and hen we have U =[B, B], he unpoen radcal of B and W 0 = N G T/T, he Weyl group of G. We consder he groups Le G = GF, K = GO, K 1 = GO 1 and T = TF. N = N G T, I = {x K : px BF 0 } and W = N/TO. We call I double Iwahor subgroup of G and W double affne Weyl group of G. We defne he characers and cocharacers of T by X = X T = HomT, G m and X = X T = HomG m, T. We denoe he se of roos of G by Φ X, he se of posve roos by Φ + and he se of smple roos by Φ +. The posve Weyl chamber P + n X s defned o be P + = {λ X : λ, α 0 for all α }. Recall ha here s a paral orderng on X : λ µ f λ µ can be wren as a nonnegave lnear combnaon of smple coroos. We denoe by ρ he half sum of posve roos. We pu X = X X and le P + be he se of all λ 1, λ X such ha λ P + and λ 1, α 0 whenever λ, α = 0 for α. We defne a paral orderng on X o be he lexcographc orderng from he rgh,.e..1 λ 1, λ µ 1, µ λ >µ, or λ = µ and λ 1 µ 1. LEMMA.. The se P + s a fundamenal doman for W 0 -acon on X, where he acon s gven by wλ 1, λ =wλ 1, wλ for w W 0 and λ 1, λ X. Proof. Gven λ 1, λ X, we can fnd w 0 W 0 such ha w 0 λ P +. Assume ha w 0 λ, α = 0 for some α. If w 0 λ 1, α 0, hen we pu w 1 = w 0. Oherwse, le σ α be he smple reflecon correspondng o α. In hs case, we pu w 1 = σ α w 0, and hen w 1 λ, α = σ α w 0 λ, α = w 0 λ, α =0, and w 1 λ 1, α = σ α w 0 λ 1, α = w 0 λ 1, α > 0.

6 1386 HENRY KIM AND KYU-HWAN LEE Connung hs process, we can fnd an elemen w of W 0 such ha wλ 1, wλ P +. Now assume ha λ 1, λ, µ 1, µ P + and wλ 1, wλ =µ 1, µ for some w W 0. Snce λ, µ P + and wλ = µ, we see ha λ = µ and w s a produc of smple reflecons fxng λ, say w = σ α1 σ αs. Then I := {α 1,..., α s } forms a roo sysem wh he Weyl group W I generaed by he smple reflecons σ α1,..., σ αs. Furhermore, λ 1, α 0 and µ 1, α 0 for all 1 s. Now a sandard argumen ells us ha wλ 1 = µ 1 acually mples λ 1 = µ 1. See, for example, [Hu]. Gven λ 1, λ X, here s a w W 0 such ha wλ 1, wλ P + by Lemma., and we defne Dλ 1, λ ={µ X :wµ, wλ P + and wµ wλ 1 }. PROPOSITION.3. The group G has he followng decomposons. 1 Bruha Decomposon [Ka] G = IwI, w W and he resulng denfcaon I\G/I W s ndependen of he choce of represenaves of elemens of W. Caran Decomposon I G = 3 Caran Decomposon II G = λ P + K 1 λ K 1. Especally, we wll use he fac ha K 1 λ K 1 = λ 1,λ P + Kλ 1 1 λ K. Kλ 1 1 λ K λ 1 λ 1,λ P + Proof. For he proof of par 1, see [Ka]. The par s a classcal resul. We can choose represenaves of W 0 o be elemens of K, and see ha W W 0 T/TO. Thus, gven w W, we can wre w = w λ 1 1 λ

7 SPHERICAL HECKE ALGEBRAS OF SL 1387 wh λ 1, λ X mang a suable choce of a represenave of w W 0 such ha w K. By Lemma., each λ 1, λ X has one and only one W 0 -conjugae n P +. Combnng hese resuls wh par 1, we oban par 3. Remar.4. The Caran Decomposon II s proved by A. N. Parshn n [Pa] for G = SL n. He also proved Bruha and Iwasawa decomposons for PGL. 3. Sphercal Hece algebras of SL. We defne sphercal Hece algebras of SL and s convoluon produc afer nvesgang he decomposon of a double cose no rgh coses. In he res of hs paper, we assume ha G = SL. We mae he denfcaon X = Z Z and he paral orderng on X corresponds o he lexcographc orderng from he rgh on Z Z. We wll denoe by he same noaon he correspondng orderng on Z Z. Then we have he denfcaon and we have D, j = P + = {, j Z Z :, j 0, 0} { { Z :,, j 0, 0} f, j 0, 0, { Z :,, j 0, 0} f, j < 0, 0. In he followng lemma, we presen an explc formula of he Caran Decomposon II for SL F. a b LEMMA 3.1. Assume ha G and le, l = mn{va, vb, vc, c d a b vd}. Then K 1 l 0 c d 0 1 l K,, l 0, 0. Namely, SL F =,l 0,0 K 1 l l K. Proof. We can chec he asseron of he Lemma usng column operaons and row operaons of marces, and we om he deal. LEMMA 3.. For, j 0, 0, we have K 1 j j K = g Kg, where he dsjon unon s over g n he followng ls.

8 1388 HENRY KIM AND KYU-HWAN LEE 1 j j, 1 j j 1 j 1 lu 0 1 j for, j, l <, j, where u U are uns belongng o a fxed se of represenaves of O/1 j l O, v 1 l 1 j u 0 1 l, for, j <, l <, j, where u U are uns belongng o a fxed se of represenaves of O/ 1 j l O. Proof. Consder elemens g, g 1 j j K and wre g = 1 j 0 a b 0 1 j, g = 1 j 0 a b c d 0 1 j c d, a b a b where, c d c d K. We see ha he condon Kg = Kg s equvalen o 3.3 c d d c 1 j O. We wre c, d c, d fc d d c 1 j a b O. Noe ha f K hen c d eher c or d s a un. Le C be he se of pars c, d O such ha eher c or d s a un. Then s easy o chec ha s an equvalence relaon on C. Thus n order o deermne dfferen coses, we need only o deermne a se of represenaves of he equvalence relaons. Assume ha c, d C. Ifc s a un hen c, d 1, d/c. We wre d/c = u for some, l, j and u U {0}. If, l, j, hen c, d 1, 1 + j+l u 1, 0. Assume ha, j, l <, j. If, l =, l and, j, l <, j, hen 1, 1 + j+l + u 1, 1 j+l u for any u U {0}. If, l =, l, hen 1, 1 + j+l + u 1, 1 j+l u f and only f u u 1 j l O. Thus f c s a un, a se of represenaves of he equvalence relaon s gven by 1 + j+l 1, 0 and 1, + 1 j+l u, where, j, l <, j and u are uns belongng o a fxed se of represenaves of O/1 j l O. The represenave 1, 0 corresponds o he marx of he

9 SPHERICAL HECKE ALGEBRAS OF SL 1389 par. We consder he oher represenaves. Through elemenary operaons, K 1 j j j+l u = K 1 j 1 j 1 lu = K 1 j 1 lu 0 1 j. So we oban he marces n he par. If c s no a un, hen d s a un. In hs case, a se of represenaves of he equvalence relaon s gven by 0, 1 and + 1 j+l, u where, j <, l <, j and u are uns belongng o a fxed se of represenaves of O/1 j l O. The represenave 0, 1 corresponds o he marx of he par. For he oher represenaves, hrough elemenary operaons, we oban K 1 j 0 u j 1 + j+l = K 1 j u 1 0 u 1 l 1 j u = K 1 l 1 j u 0 1 l. So we ge he marces n he par v. Remar 3.4. If j l, hen he cardnaly of O/ O s uncounable. Hence n general, he double cose K 1 j j K s an uncounable unon of rgh coses of K. Ths s very dfferen from he classcal p-adc case of SL Q p where any double cose of SL Z p s a fne unon of rgh coses of SL Z p. Now we begn our consrucon of Hece algebras. 1 j l Defnon 3.5. A C-valued funcon f on G s called sphercal f f sasfes he followng properes: f 1 x =fx for 1, K and x G, here exss, j P + such ha f x =0 fx / m D,j K m 1 j m j K. If f s sphercal, hen we can always fnd he mnmal, j P + sasfyng 3.6 wh respec o he orderng.1 on X. The mnmal, j wll be called he wegh of f.

10 1390 HENRY KIM AND KYU-HWAN LEE For each, j 0, 0, we le χ,j be he characersc funcon of he double cose K 1 j j K. Then a sphercal funcon of wegh, j can be wren as 3.7 a χ,j for some, j 0, 0 and a C. Defnon 3.8. We defne he convoluon produc of wo characersc funcons by χ,j τxz 1 m f x K 1 j+l 0 χ,j χ,l x = z 0 m m D+,j+l 1 j l K, 0 oherwse, 3.9 m where τx = 1 j+l 0 m 0 1 m j l for x K 1 j+l m j l K, and he sum s over he represenaves z of he decomposon K 1 l l K = z Even hough a double cose of K s generally an uncounable unon of rgh coses of K, we show n he followng lemma ha gven x, here are only fnely many nonzero erms n 3.9. Hence he convoluon produc s well-defned. Now we prove: PROPOSITION The convoluon produc χ,j χ,l s a sphercal funcon. Moreover, we have he followng explc formulas: 1 If > 0, hen χ,0 χ,0 = χ, q If,0<, l, hen 0<r< Kz. q r χ r,0 + q 1+ 1 χ 0,0. q χ,0 χ,l = χ,l χ,0 = χ +,l q r χ + r,l + q χ +,l. q 0<r<

11 3 If j > 0 and l > 0, hen Proof. We may le SPHERICAL HECKE ALGEBRAS OF SL 1391 χ,j χ,l = χ +,j+l q r χ + r,j+l. q r>0 m x = τx = 1 j+l m j l, m D +, j + l. We consder K 1 l l K = z Kz, where he dsjon unon s over z n he ls -v of Lemma 3.. We need o deermne he condons under whch χ,j xz 1 =1. Snce he oher cases are smlar, we prove only he par 3 of he lemma. So we assume ha j > 0 and l > 0. In hs case, m D +, j + l f and only f m +. If z = 1 l l hen we have χ,j xz 1 m+ =χ 1 j+l 0,j 0 1 m j l for all m +. If z = 1 l l hen we have =0 χ,j xz 1 m =χ 1 j 0,j 0 1 m+ j =1 m = +. If z = 1 l 1 l u 0 1 l for, l, l <, l and u U belongng o a fxed se of represenaves of O/1 l l O, hen we ge for all m +. χ,j xz 1 m+ =χ 1 j+l 1 m+ j+l+l u,j 0 1 m j l =0

12 139 HENRY KIM AND KYU-HWAN LEE v z = 1 l 1 l u 0 1 l for, l <, l <, l and u U belongng o a fxed se of represenaves of O/ 1 l l O, hen χ,j xz 1 m =χ 1 j+l l 1 m j u,j 0 m+ l = l, m = + and <. 1 j l+l =1 Noe ha for each < he number of dfferen u s s 1 1 q q. Snce here s only fne number of z s for whch χ,j xz 1 = 0, we have χ,j χ,l x = z 1 f m = +, χ,j xz 1 = 1 1 q q f m = + for each <, 0 oherwse. Therefore, χ,j χ,l = χ +,j+l q χ + q,j+l < = χ +,j+l q r χ + r,j+l. q r>0 For example, we have he followng denes: 3.11 χ 1,0 χ 1,0 = χ,0 +q 1χ 1,0 +q + qχ 0,0, χ 1,0 χ 0,1 = χ 1,1 +q 1χ 0,1 + q χ 1,1, χ 0,1 χ 0,1 = χ 0, +1 1 q >0 q χ,. Assume ha f = p a pχ p,j and g = q b qχ q,l are sphercal funcons. We defne he convoluon produc f g by 3.1 f g = a p χ p,j b q χ q,l = a p b q χ p,j χ q,l. p q I follows from Proposon 3.10 ha we can wre f g = r + c r χ r,j+l, p, q and noe ha χ r,j+l appears n he expanson of χ p,j χ q,l only f p + q r. Snce he number of such pars p, q s fne, he coeffcen c r s well defned for each

13 SPHERICAL HECKE ALGEBRAS OF SL 1393 r +. Therefore he convoluon produc f g s a well defned sphercal funcon of wegh +, j + l. THEOREM The convoluon produc s commuave. Namely, f g = g f for sphercal funcons f and g on G. Proof. Noe ha we have χ,j χ,l = χ,l χ,j n he par 3 of Proposon Wh hs observaon, he proposon follows from Proposon 3.10 and defnons. Now we are ready o nroduce he man objec of hs paper. Defnon We defne he sphercal Hece algebra HG, K of G relave o K o be he C-algebra whose elemens are fne lnear combnaons of sphercal funcons on G wh he mulplcaon gven by he convoluon produc 3.9 and 3.1. PROPOSITION a The elemens χ,0, > 0 and χ,1, Z are conaned n he subalgebra generaed by χ 1,0, χ 0,1 and χ 1,1. b The elemen χ,j for each, j, j> 1 s gven by he formula χ,j = χ,j 1 q 1 χ r,j 1 χ 0,1. r>0 Proof. Usng Proposon 3.10, one can chec he asserons and we om he deal. Remar If we allow nfne sums of he form 3.7, an nducon argumen usng he above proposon shows ha he algebra HG, K s generaed by hree elemens χ 1,0, χ 0,1 and χ 1,1. 4. Sphercal Hece algebra of T. In hs secon, we consruc he sphercal Hece algebra of T relave o TO for G = SL F. Defnon 4.1. A C-valued funcons f on T s called sphercal f f sasfes he followng condons: 1 f xz =f x for z TO and x T, here exs, j X such ha 4. f x =0 fx / m D,j m 1 j m j TO. We can fnd, j X sasfyng 4., whch s mnmal n he sense ha f anoher, j sasfes he condon hen D, j. A mnmal, j s unquely

14 1394 HENRY KIM AND KYU-HWAN LEE deermned, and wll be called he wegh of f. For each, j Z Z, we defne c,j o be he characersc funcon of he double cose 1 j j TO. Then a sphercal funcon on T can be wren as one of he followng: 4.3 a c,j, a c, j, 0 a c,0, 0 Defnon 4.4. We defne he convoluon produc c,j c,l by a c,0 a C, j > c,j c,l = { c+,j+l f jl 0, 0 oherwse. Assume ha f = p a pc p,j and g = q b qc q,l are sphercal funcons of T of weghs, j,, l, resp. where j, l > 0. We defne he convoluon produc f g by 4.6 f g = a p c p,j b q c q,l = a p b q c p,j c q,l. p q p, q Then f g = p, q a p b q c p+q,j+l = r + p,q r=p+q a p b q c r,j+l. Hence f g s a well-defned sphercal funcon of wegh +, j + l. If f, g are of he forms a c, j, 0 a c,0,or 0 a c,0, hen f g s defned n a smlar way. Noe ha f f, g have weghs, j,, l resp. where jl < 0, hen by 4.5 he convoluon produc f g s dencally zero. Defnon 4.7. We defne he sphercal Hece algebra HT, TO of T o be he C-algebra whose elemens are fne lnear combnaons of sphercal funcons on T wh he mulplcaon gven by he convoluon produc 4.5 and 4.6. We need he noon of W 0 -nvaran subalgebra of HT, TO; he algebra HT, TO W 0 s defned o be he subalgebra of HT, TO conssng of he elemens f sasfyng he condon f wxw 1 =f x for w W 0.

15 SPHERICAL HECKE ALGEBRAS OF SL 1395 Remar 4.8. If we allow nfne sums of he forms n 4.3, he W 0 -nvaran subalgebra HT, TO W 0 s generaed by hree elemens c 1,0 +c 1,0, c 0,1 +c 0, 1 and c 1,1 + c 1, Saae somorphsm. In hs secon, we defne he Saae somorphsm for SL, namely, he somorphsm beween HG, K and HT, TO W 0. For he classcal p-adc groups, I. Saae [Sa] consruced such somorphsm. We follow hs consrucon, usng he measure and negraon n Secon 1. We hope o generalze our resuls o arbrary spl sem-smple algebrac groups Defnon 5.1. A group homomorphsm δ : T RX s defned o be δx =q X j We denfy F wh UF hrough for x 1 j j TO. u : F UF, ua = 1 a. 0 1 Le dµa be he nvaran measure on UF, gven by he nvaran measure on F as n secon 1 and he somorphsm u. Defnon 5.4. We defne a map S : HG, K HT, TO by S f x = δx 1 X j F f xuadµa, f x Z 0 oherwse, 1 ±j j TO, 5.5 for f whose wegh s, j P +, and by exendng o he whole HG, K hrough lneary. I s clear ha S f sto-nvaran. However, s no clear a pror ha S f sac-valued funcon snce he measure aes value n RX. We show n he nex wo proposons ha S f sac-valued funcon and he map S s a well-defned map. and 5.7 PROPOSITION 5.6. { } Sχ,j =q c,j + c, j +1 1 q,j + c, j for, j > 0, 0, <c Sχ,0 =q c,0 + c, q c,0 for > 0. <<

16 1396 HENRY KIM AND KYU-HWAN LEE Proof. From he defnon, we need only o consder x 1 j 0 0 Z 1 j TO 1 j j TO. Snce S f sto-nvaran, s enough o consder x = 1 l l, Z and l = ±j. From he defnon, we ge δx 1 1 a = q X l.ifua = 0 1 UF, hen xua = 1 l a1 l 0. 1 l We frs consder he case, l 0, 0,.e., l = j. Ifva, j, hen follows from Lemma 3.1 ha xua K 1 j j K; oherwse, a xua K 1 1 j 0 0 a1 j K. We oban Sχ,j x =q X j χ,j xuadµa F = q X j χ 1 j 0,j va, j 0 1 j dµa + q X j a χ 1 1 j 0,j va<, j 0 a1 j dµa. In he second negral, noe ha va 1 >, j. So f, j =, j, hen he second negral s zero. Also {a F va, j} = 1 j O n he noaon of Defnon 1.. Hence va, j dµa =q X j. Therefore f, j =, j, 1 j Sχ,j x =q X j q X j = q. If, j >, j,.e., <, hen he frs negral s zero. The second negral s nonzero only when va =, j. Here {a F va =, j} = 1 j U and he measure of U s 1 1 q. Hence va=, j dµa =q+ X j 1 1 q. Therefore, f, j >, j, Sχ,j x =q X j q + X j 1 1 = q 1 1. q q

17 SPHERICAL HECKE ALGEBRAS OF SL 1397 Nex we consder he case, l < 0, 0. A smlar calculaon as n he above gves us { q f, j =, l, Sχ,j x = q 1 1 q f, j >, l. Combnng hese resuls we oban he equales n 5.7. PROPOSITION 5.8. For each, j 0, 0, S a χ,j = a Sχ,j. Proof. I s enough o prove ha he rgh hand sde s well defned. From 5.7, we only need o chec he well defnedness of he double sum a q c l,j + c l, j. l< I s equal o c l,j + c l, j l Here =l a q s a fne sum and well defned. a q. Hence, by Proposon 5.6 and Proposon 5.8, he map S s well defned and he mage of he map S s conaned n he W 0 -nvaran subalgebra HT, TO W 0. LEMMA 5.9. We have =l Sχ,j χ,l =Sχ,j Sχ,l for, j 0, 0 and, l 0, 0. Proof. Usng Proposon 3.10, Proposon 5.6 and Proposon 5.8, one can sraghforwardly chec he deny and we om he deal. THEOREM Assume ha G = SL. Then he map S : HG, K HT, TO W 0 s an algebra somorphsm. Proof. In he formula 5.7 we see ha he maxmal wegh of he erms n he rgh-hand sde s he same as he wegh of χ,j. Thus S s wegh-preservng and so njecve. Snce an elemen of HT, TO W 0 s a fne lnear combnaon of he elemens of he form 5.11 g = a c,j + c, j, a C,, j 0,

18 1398 HENRY KIM AND KYU-HWAN LEE we need only o fnd an elemen f of HG, K such ha S f =g. Ifj = 0, hen 5.11 s a fne sum and s easy o fnd such an f usng 5.7. So we assume ha j > 0. By Proposon 5.8 and 5.7, we oban S χ,j q 1 χ r,j = q c,j + c, j. r>0 Thus we pu 5.1 f = q a χ,j q 1 χ r,j. r>0 If we rewre 5.1 as f = b χ,j, hen each b C,, s well defned. Now Proposon 5.8 enables us o have S f =g. Hence, S s surjecve. I follows from Proposon 5.8 and Lemma 5.9 ha S s an algebra homomorphsm. Therefore, S s an algebra somorphsm. Remar We expec n he near fuure See [K-L] o consruc an nvaran RX-valued measure dγ on GF for G a conneced spl sem-smple algebrac group. We follow [Go], namely, we defne an addve nvaran measure on he produc space F n and oban he ransformaon rule. Then usng he bg cell decomposon, we can exend o an nvaran measure on GF. Then we can defne he convoluon produc of wo sphercal funcons f and g on GF by G f xy 1 gydγy f x f gx = 0 oherwse. µ Dλ 1 +µ 1,λ +µ Kµ 1 λ + µ K, The defnon of he convoluon produc 3.9 was movaed by hs formula. We need o defne he convoluon produc n hs way so ha gven x, he funcon y f xy 1 gy s a fne lnear combnaon of characersc funcons of rgh coses of K. Then f g becomes a C-valued funcon, even hough he measure dγ aes values n RX. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TORONTO, TORONTO, ON M5S 3G3, CANADA E-mal: henrym@mah.orono.edu, hlee@mah.orono.edu

19 SPHERICAL HECKE ALGEBRAS OF SL 1399 REFERENCES [Ca] P. Carer, Represenaons of p-adc groups: a survey, Auomorphc Forms, Represenaons and L- funcons, Par 1, Proc. Sympos. Pure Mah., vol. 33, Amer. Mah. Soc., Provdence, RI, 1979, pp [Fe] I. Feseno, Analyss on arhmec schemes I, Doc. Mah. Exra Volume: Kazuya Kao s Ffeh Brhday 003, [Fe1], Measure, negraon and elemens of harmonc analyss on generalzed loop spaces, preprn, avalable a [LF] I. Feseno and M. Kurhara eds., Invaon o Hgher Local Felds, Geomery & Topology Monographs, vol. 3, Geomery & Topology Publcaons, Covenry, 000. [Go] J. Gordon, Movc Haar measure on reducve groups, preprn, alg-geom/ [Hu] J. E. Humphreys, Reflecon Groups and Coxeer Groups, Cambrdge Sud. Adv. Mah., vol. 9, Cambrdge Unversy Press, Cambrdge, [Ka] M. Kapranov, Double affne Hece algebras and -dmensonal local felds, J. Amer. Mah. Soc , [K-L] H. Km and K.-H. Lee, n preparaon [Pa] A. N. Parshn, Vecor bundles and arhmec groups I, Trudy Ma. Ins. Selov , Teor. Chsel, Algebra Algebr. Geom., preprn alg-geom/ v1. [Sa] I. Saae, Theory of sphercal funcons on reducve algebrac groups over p-adc felds, Ins. Haues Éudes Sc. Publ. Mah , 5 69.