Katz-Radon transform of l-adic representations

KATZ-RADON TRANSFORM OF `-ADIC REPRESENTATIONS ANTONIO ROJAS-LE ´ ON Abs ac . We p o e a simple explici o mula o he local Ka z-Radon ans- o m o an `-adic ep esen a ion o he Galois g oup o he ac ion ield o a s ic ly henselian disc e e alua ion ing wi h posi i e esidual cha ac e is ic, which can be de ined as he local addi i e con olu ion wi h a ixed ame cha - ac e . The o mula is simila o one p o ed by D. A inkin in he D-module se ing, and answe s a ques ion posed by N. Ka z. 1. In oduc ion In [10, 3.4.1], N. Ka z de ines some unc o s on he ca ego y o con inuous `-adic ep esen a ions o he ine ia g oups I0and I∞o he p ojec i e line o e ¯ ka 0 and in ini y, whe e ¯ kis he algeb aic closu e o a ini e ield o cha ac e is ic p and `is a p ime di e en om p. These unc o s a ise du ing his s udy o middle con olu ion o shea es on he a ine line and, oughly speaking, co espond o locally con ol ing a ep esen a ion wi h a ixed ame cha ac e Lχo I0o I∞. They a e de ined using G. Laumon’s local Fou ie ans o m unc o s, and in ac co espond o aking he enso p oduc wi h he conjuga e ame cha ac e L¯χon he o he side o he equi alence o ca ego ies gi en by hese unc o s. Ka z asks [10, 3.4.1] whe he he e is a simple exp ession o he unc o s de ined in his way. Recen ly, D. A inkin [1] has s udied he analog o Ka z’s unc o in D-module heo y: i Kis a ield o cha ac e is ic 0, K((x)) is he ield o Lau en se ies o e Kand Dx he ing o di e en ial ope a o s wi h coe icien s in K((x)), he local Ka z-Radon ans o m o a gi en λ∈K−Zis an equi alence o ca ego ies ρλ:Dx-mod→ Dx-mod, o iginally de ined in [3]. A inkin p o es he simple o mula [1, Theo em C] ρλ(F)∼ =F ⊗ Kλ(a+1) o any F ∈ Dx-mod wi h a single slope a, whe e Kµis he Kumme Dx-module o ank 1 gene a ed by e, on which he de i a i e ac s by d dxe=µ xe. In his a icle we will p o e a simila o mula in he `-adic case. Mo e p ecisely, o a ixed ame `-adic cha ac e Lχand an `-adic ep esen a ion Fo I0, le ρχ(F) := FTψ,−1 (0,∞)(L¯χ⊗FTψ (0,∞)F) whe e FTψ (0,∞)deno es Laumon’s local Fou ie ans o m unc o . I Fhas a single slope a=c/d (wi h c, d ela i ely p ime posi i e in ege s), we will p o e ha he e is an isomo phism o I0- ep esen a ions ρχ(F)∼ =F ⊗ L⊗(a+1) χ Ma hema ics Subjec Classi ica ion: 14F20,11F85,11S99 Pa ially suppo ed by P08-FQM-03894 (Jun a de Andaluc´ıa), MTM2010-19298 and FEDER. 1 2 ANTONIO ROJAS-LE ´ ON whe e L⊗(a+1) χis any d- h oo o he cha ac e L⊗(c+d) χ. Fo a la ge class o ep esen a ions Fo I0(in pa icula o many o hose who appea in applica ions), he isomo phism can be p o en ia he explici o mulas o he local Fou ie ans o ms gi en by L. Fu [5] and A. Abbes and T. Sai o [2]. In his a icle we ake a di e en app oach ha wo ks o any F, and is independen o any explici exp ession o he local Fou ie ans o ms. 2. The Ka z-Radon ans o m Fix a ini e ield ko cha ac e is ic p > 0 and an algeb aic closu e ¯ k. Le P1 ¯ k be he p ojec i e line o e ¯ kand, o e e y ∈P1(¯ k) = ¯ k∪ {∞}, deno e by I i s ine ia g oup a : o 6=∞, i x− deno es a local coo dina e a , i is he Galois g oup o he ac ion ield o he henseliza ion o he local ing ¯ k[x](x− ). We ha e an exac sequence [8, 1.0] 0→P →I →Y `6=p Z`(1) →0 o e e y ∈P1(¯ k), whe e P is he only p-Sylow subg oup o I . Mo eo e , he e is a canonical il a ion o I by he highe ami ica ion g oups I( ) ⊇I(s) o 0 ≤ < s ∈R which a e no mal in I . Fix a p ime `6=p, and deno e by R he abelian ca ego y o con inuous `-adic ep esen a ions o I (i.e. con inuous ep esen a ions F:I →GLn(¯ Q`), whose image is in GLn(Eλ) o some ini e ex ension Eλo Q`). Fo e e y i educible F ∈ R , he slope o Fis in { ≥0|F|I( ) is i ial}. I is a non-nega i e a ional numbe . In gene al, he slopes o Fa e he slopes o he i educible componen s o F. Fo e e y F he e is a canonical di ec sum decomposi ion [8, Lemma 1.8] (1) F∼ =M ≥0 F wi h F ha ing a single slope . The slope 0 ( ame) pa will be deno ed by F .F is said o be ame ( espec i ely o ally wild) i F=F ( esp. F = 0). Fo e e y ≥0 le R deno e he ull subca ego y o R consis ing o ep esen- a ions wi h a single slope . We ha e a decomposi ion R =M ≥0 R in he sense ha e e y F ∈ R has a decomposi ion (1) and HomR (F,G) = 0 i F ∈ R ,G ∈ Rs and 6=s[8, P oposi ion 1.1]. Le k0⊆¯ kbe a ini e ex ension o k, and χ:k0∗ →¯ Q∗ `a mul iplica i e cha ac e . By [4, 1.4-1.8] he e is an associa ed smoo h Kumme shea Lχon Gm,¯ k, which is a ame cha ac e o I0(and o I∞) o he same o de as χ. I k0⊆k00 is ano he ex ension, he shea es de ined by χand χ◦Nmk00 /k0:k00∗ →¯ Q∗ `a e isomo phic. Mo eo e , e e y ame cha ac e o I0(and o I∞) can be ob ained in his way. Whene e we speak abou a ame cha ac e o I0, we will implici ly assume ha we ha e made a choice o such a ini e ex ension o kand o a cha ac e . KATZ-RADON TRANSFORM OF `-ADIC REPRESENTATIONS 3 Fix a non- i ial addi i e cha ac e ψ:k→¯ Q∗ `. The local Fou ie ans o m unc o s, de ined by G. Laumon in [11], gi e equi alences o ca ego ies FTψ (0,∞):R0→ R<1 ∞, FTψ (∞,∞):R>1 ∞→ R>1 ∞ and FTψ (∞,0) :R<1 ∞→ R0 (whe e R<1 ∞=L <1R ∞and R>1 ∞=L >1R ∞) ha desc ibe he ela ionship be ween he local monod omies o an `-adic shea on A1 ¯ kand i s Fou ie ans o m wi h espec o ψ. The Ka z-Radon ans o m is de ined in e ms o hem. De ini ion 2.1. Fix a ame cha ac e Lχo I0. The (local) Ka z-Radon ans o m (wi h espec o Lχ) is he unc o ρχ:R0→ R0gi en by ρχ(F) = FTψ,−1 (0,∞)(FTψ (0,∞)Lχ⊗FTψ (0,∞)F) = FTψ,−1 (0,∞)(L¯χ⊗FTψ (0,∞)F). The Ka z-Radon ans o m is an au o-equi alence o he ca ego y R0(since i is a composi ion o h ee equi alences o ca ego ies). I p ese es dimensions and slopes, and o ame Fi is gi en by ρχ(F) = F ⊗ Lχ[10, 3.4.1]. Fo o ally wild F, i can be in e p e ed as he “local addi i e con olu ion” o Fand Lχ[10, 3.4.3]: i we ex end F o a smoo h shea on Gm,¯ k, amely ami ied a in ini y, hen ρχ(F) is he wild pa o he local monod omy a 0 o F ∗ Lχ, whe e F ∗ Lχ= R1σ!(FLχ) and σ:A2 ¯ k→A1 ¯ kdeno es he addi ion map (in [10], he “middle con olu ion” is used ins ead, bu ha one di e s om he one used he e only by A in-Sh eie com- ponen s, which a e smoo h a 0 and he e o e do no a ec he local monod omy). No ice ha , in pa icula , ρχis independen o he choice o he addi i e cha ac e ψ. Mo e in insically, i can be desc ibed in e ms o anishing cycles unc o s [11, 2.7.2]: I X=A2 (0,0) ( espec i ely S=A1 (0)) deno es he henseliza ion o A2 ¯ ka (0,0) ( esp. he henseliza ion o A1 ¯ ka 0) hen ρχ(F)∼ =R1Φ(σ, FLχ)(0,0), whe e RΦ(σ, FLχ) is he anishing cycles complex o he addi ion map σ:X→S wi h espec o he shea FLχon X. Simila ly, i also has an in e p e a ion as a “local mul iplica i e con olu ion” [12, Co olla y 5.6]: I X=G2 m,(1,1) ( espec i ely S=Gm,(1)) deno es he henseliza ion o Gm,¯ ka (1,1) ( esp. he henseliza ion o Gm,¯ ka 1) hen ρχ(F)∼ =R1Φ(µ, F Lχ)(1,1), whe e RΦ(µ, FLχ) is he anishing cycles complex o he mul iplica ion map µ:X→Swi h espec o he shea FLχon X, and Fand Lχa e iewed as ep esen a ions o I1 ia he isomo phism I0∼ =I1 ha maps he uni o mize x a 0 o he uni o mize x−1 a 1. The main esul o his a icle is he ollowing simple exp ession o ρχ: Theo em 2.2. Le F ∈ R0be o ally wild wi h a single slope a > 0. W i e a=c/d, whe e cand da e ela i ely p ime posi i e in ege s. Le Lηbe any ame cha ac e o I0such ha L⊗d η=L⊗(c+d) χ. Then ρχ(F)∼ =F ⊗ Lη. 4 ANTONIO ROJAS-LE ´ ON In o he wo ds, we ha e he o mula (2) ρχ(F)∼ =F ⊗ L⊗(a+1) χ whe e L⊗(a+1) χs ands o “any cha ac e ha can easonably be called L⊗(a+1) χ”. By he decomposi ion R0=L ≥0R 0, his de e mines ρχ(F) o any F ∈ R0, hus answe ing he ques ion posed by N. Ka z in [10, 3.4.1]. A ques ion ha emains open is he ollowing: in he a icle we p o e ha ρχ(F)∼ =F ⊗ Lη, independen ly o any Fwi h slope a. So he unc o s Ra 0→ Ra 0 gi en by ρχand (−)⊗ Lηmap any F o isomo phic objec s. Is he e an ac ual isomo phism o unc o s be ween hem? In he a i ma i e case, is he e a simple way o cons uc i ? 3. P oo o he main heo em In his sec ion we will p o e heo em 2.2. We will s a wi h he case whe e F ∈ R0is i educible. Lemma 3.1. Le F ∈ R0. Then F 6= 0 i and only i he e exis s  > 0such ha o e e y G ∈ R0wi h a single slope b∈(0, )we ha e Swan(F ⊗ G)>Swan(F) dim(G). P oo . Suppose ha F 6= 0, and le a0= 0 < a1<· · · < a be he slopes o F, wi h mul iplici ies n0, n1, . . . , n . Then Swan(F) = Pniai. Le =a1. Then o e e y G ∈ R0wi h a single slope b∈(0, ) he enso p oduc F ⊗ G has slopes b<a1<· · · < a wi h mul iplici ies n0m, n1m, . . . , n mwhe e m= dim(G) by [8, Lemma 1.3]. The e o e Swan(F ⊗ G) = n0mb + X i=1 nimai> X i=1 nimai= Swan(F) dim(G). Con e sely, suppose ha F = 0, and le a1<· · · < a be he slopes o F. Then o e e y G ∈ R0wi h a single slope b∈(0, a1) he enso p oduc F ⊗ G has he same slopes as Fby [8, Lemma 1.3], and in pa icula Swan(F ⊗ G) = Swan(F) dim(G). This p o es he lemma, since o e e y  > 0 he e exis ep esen a ions in R0wi h slope b∈(0, ) ( o ins ance, one may ake [n]∗H, whe e H ∈ R0has slope a > 0 and nis a p ime o pin ege g ea e han a/ [8, 1.13.2]).  Fo any wo objec s K, L ∈ Db c(A1 ¯ k,¯ Q`), we will deno e by K∗L∈ Db c(A1 ¯ k,¯ Q`) hei addi i e con olu ion: K∗L= Rσ!(KL) whe e σ:A2 ¯ k→A1 ¯ kis he addi ion map. Lemma 3.2. Le K, L, M ∈ Db c(A1 ¯ k,¯ Q`). Then RΓc(A1 ¯ k,(K∗L)⊗M)∼ =RΓc(A1 ¯ k, K ⊗((τ∗ −1L)∗M)) whe e τ−1:A1 ¯ k→A1 ¯ kis he addi i e in e sion. P oo . We ha e RΓc(A1 ¯ k,(K∗L)⊗M) = RΓc(A1 ¯ k,Rσ!(KL)⊗M) = = RΓc(A1 ¯ k,Rσ!((KL)⊗σ∗M)) = RΓc(A2 ¯ k,(KL)⊗σ∗M) KATZ-RADON TRANSFORM OF `-ADIC REPRESENTATIONS 5 by he p ojec ion o mula. I π1, π2:A2 ¯ k→A1 ¯ ka e he p ojec ions hen RΓc(A2 ¯ k,(KL)⊗σ∗M) = RΓc(A2 ¯ k, π∗ 1K⊗π∗ 2L⊗σ∗M). Conside he au omo phism φ:A2 ¯ k→A2 ¯ kgi en by (x, y)7→ (x+y, −y). Then σ=π1◦φ,π1=σ◦φand τ−1◦π2=π2◦φ. I ollows ha RΓc(A2 ¯ k, π∗ 1K⊗π∗ 2L⊗σ∗M)∼ =RΓc(A2 ¯ k, φ∗π∗ 1K⊗φ∗π∗ 2L⊗φ∗σ∗M) = = RΓc(A2 ¯ k, σ∗K⊗π∗ 2τ∗ −1L⊗π∗ 1M) = RΓc(A1 ¯ k,Rσ!(σ∗K⊗π∗ 2τ∗ −1L⊗π∗ 1M)) ∼ = ∼ =RΓc(A1 ¯ k, K ⊗Rσ!((τ∗ −1L)M)) = RΓc(A1 ¯ k, K ⊗((τ∗ −1L)∗M)).  I Fis a smoo h ¯ Q`-shea on Gm,¯ kwhich is o ally wild a 0, hen o e e y ∈¯ k he shea F ⊗Lχ( −x)(ex ended by ze o o A1 ¯ k) is o ally wild a 0 and has no punc ual sec ions (whe e Lχ( −x)is he pull-back o he Kumme shea Lχunde he map x7→ −x), so i s only non-ze o cohomology g oup wi h compac suppo is H1 c. We conclude ha he only non-ze o cohomology shea o F[0] ∗ Lχ[0] ∈ Db c(A1 ¯ k,¯ Q`) is H1= R1σ!(F ⊗ Lχ). We will deno e his shea by F ∗ Lχ. Lemma 3.3. Le F,G ∈ R0. Then Swan(ρχ(F)⊗ G) = Swan(F ⊗ G). P oo . By addi i i y o he Swan conduc o , we may assume ha Fis i educible, and in pa icula ha i has a single slope a≥0. I a= 0 hen ρχ(F)∼ =F ⊗ Lχ, so he equali y is clea . Suppose ha a > 0. By [7, Theo em 1.5.6], Fand Gcan be ex ended o smoo h shea es on Gm,¯ k, amely ami ied a in ini y, which we will also deno e by Fand G. Le Fand Gbe also hei ex ensions by ze o o A1 ¯ k. Using he compa ibili y be ween Fou ie ans o m wi h espec o ψand con- olu ion [11, P oposi ion 1.2.2.7], we ha e F ∗ Lχ= FT ¯ ψ(FTψF ⊗ FTψLχ) = FT ¯ ψ(FTψF ⊗ L¯χ), whe e FTψFdeno es he “nai e” Fou ie ans o m in he sense o [8, 8.2], ha is, he (−1)- h cohomology shea o he Fou ie ans o m o F[1] ∈ Db c(A1 ¯ k,¯ Q`) (which is i s only non-ze o cohomology shea , since Fis o ally wild a ze o and he e o e i is Fou ie [8, Lemma 8.3.1]). Le nbe he ank o F, and deno e by F(∞)∈ R∞i s local monod omy a in ini y, which is a ame ep esen a ion o I∞. By Ogg-Sha a e ic [6, Expos´e X, Co ollai e 7.12], FTψFis smoo h on Gm,¯ ko ank na+n=n(a+1). By Laumon’s local Fou ie ans o m heo y [9, Theo em 13], FTψFhas a single slope a a+1 a in ini y, wi h mul iplici y n(a+ 1), and i s monod omy a 0 has a i ial pa o dimension na and i s quo ien is he dual [ F(∞)o F(∞). Then FTψF ⊗L¯χalso has a single slope a a+1 a in ini y wi h mul iplici y n(a+ 1), and i s monod omy Ma 0 si s in an exac sequence (3) 0 → L⊕na ¯χ→M→ [ F(∞)⊗ L¯χ→0. I s in e se Fou ie ans o m, by Ogg-Sha a e ic, is smoo h o ank n(a+1) on Gm,¯ k, and by local Fou ie ans o m i s wild pa a 0 has slope awi h mul iplici y n. 6 ANTONIO ROJAS-LE ´ ON In ac , his wild pa is simply ρχ(F) by he addi i e con olu ion in e p e a ion o ρχ. I s monod omy a in ini y si s in an exac sequence (4) 0 → L⊕na χ→(F ∗ Lχ)(∞)→ F(∞)⊗ Lχ→0 ob ained om (3) by local Fou ie ans o m. So F ∗ Lχhas ank n(a+ 1) on Gm,¯ k, and i s monod omy a 0 is he di ec sum o ρχ(F) and a cons an pa o dimension na = Swan(F). So Swan0((F ∗ Lχ)⊗ G) = Swan(ρχ(F)⊗ G) + Swan(F)Swan(G). In pa icula , by Ogg-Sha a e ic, he Eule cha ac e is ic o he shea (F ∗ Lχ)⊗ G (ex ended by ze o o A1 ¯ k) is −Swan(ρχ(F)⊗ G)−Swan(F)Swan(G). Using lemma 3.2, p ope base change, and he ac ha χ(Gm,¯ k, K ⊗ Lχ) = χ(Gm,¯ k, K) o any objec K∈ Db c(Gm,¯ k,¯ Q`), we ge Swan(ρχ(F)⊗ G) + Swan(F)Swan(G) = χ(A1 ¯ k,(F[1] ∗ Lχ[1]) ⊗ G) = =χ(A1 ¯ k,Lχ⊗(τ∗ −1F[1] ∗ G[1])) = χ(Gm,¯ k, τ∗ −1F[1] ∗ G[1]) = =χ(A1 ¯ k, τ∗ −1F[1] ∗ G[1]) − ank0(τ∗ −1F[1] ∗ G[1]) = =χ(A1 ¯ k,F[1])χ(A1 ¯ k,G[1]) −χ(A1 ¯ k,F[1] ⊗ G[1]) = = Swan(F)Swan(G) + Swan(F ⊗ G) whe e ank0o a de i ed ca ego y objec deno es he al e na ing sum o he anks a 0 o i s cohomology shea es, so Swan(ρχ(F)⊗ G) = Swan(F ⊗ G).  P oposi ion 3.4. Le F ∈ R0be o ally wild and i educible. Then he e exis s a ame cha ac e Lηo I0such ha ρχ(F)∼ =F ⊗ Lη. P oo . Le b Fbe he dual ep esen a ion. We claim ha he ame pa o ρχ(F)⊗b F is non-ze o. By lemma 3.1, i su ices o show ha he e is an  > 0 such ha , o any G ∈ R0wi h slope b∈(0, ), Swan(ρχ(F)⊗b F ⊗G)>Swan(ρχ(F)⊗b F) dim(G). Bu by lemma 3.3, we ha e Swan(ρχ(F)⊗b F ⊗ G) = Swan(F ⊗ b F ⊗ G) and Swan(ρχ(F)⊗b F) = Swan(F ⊗ b F) and, since b Fis he dual o F, he enso p oduc F ⊗ b Fhas a i ial quo ien and, in pa icula , has non- i ial ame pa . By lemma 3.1, he e exis s  > 0 such ha , o any G ∈ R0wi h slope b∈(0, ), Swan(F ⊗ b F ⊗ G)>Swan(F ⊗ b F) dim(G). Since he ame pa o ρχ(F)⊗b Fis non-ze o and i is a di ec summand, i con ains a ame cha ac e Lηo I0as a sub ep esen a ion. Then ρχ(F)⊗b F ⊗ L¯η=ρχ(F)⊗ F ⊗ Lη= Hom(F ⊗ Lη, ρχ(F)) con ains a i ial sub ep esen a ion, so HomI0(F ⊗ Lη, ρχ(F)) 6= 0. Since bo h ρχ(F) and F ⊗Lηa e i educible, any non-ze o I0-equi a ian map F ⊗Lη→ρχ(F) mus be an isomo phism.  P oposi ion 3.5. Le F ∈ R0be o ally wild and i educible o dimension nand slope a, and le Lηbe a ame cha ac e o I0such ha ρχ(F)∼ =F ⊗ Lη. Then L⊗n η∼ =L⊗n(a+1) χ. KATZ-RADON TRANSFORM OF `-ADIC REPRESENTATIONS 7 P oo . Ex end F o a smoo h `-adic shea on Gm,¯ k, amely ami ied a in ini y, also deno ed by F. Le Falso deno e i s ex ension by ze o o A1 ¯ k. By he p oo o lemma 3.3, he shea F ∗ Lχis smoo h on Gm,¯ k, i s monod omy a 0 is he di ec sum o ρχ(F)∼ =F ⊗ Lηand a i ial pa o dimension na, and i s monod omy a in ini y si s in he exac sequence (4). I s de e minan is hen a smoo h shea o ank 1 on Gm,¯ k, whose monod omy a 0 is de (F)⊗ L⊗n η, and whose monod omy a ∞is de (F(∞))⊗ L⊗n(a+1) χ. Then de (F)⊗ L⊗n ¯η⊗de (F ∗ Lχ) is a ank 1 smoo h shea on Gm,¯ k, wi h i ial monod omy a 0 and amely ami ied a in ini y. Since he ame undamen al g oup o A1 ¯ kis i ial, we conclude ha de (F ∗ Lχ)∼ =de (F)⊗ L⊗n η as shea es on Gm,¯ k. Compa ing hei monod omies a in ini y gi es he desi ed isomo phism.  I emains o show ha any such Lηwo ks. Lemma 3.6. Le F ∈ R0be i educible o dimension n, and le Lηbe a ame cha ac e o I0such ha L⊗n ηis i ial. Then F ⊗ Lη∼ =F. P oo . W i e n=n0pα, whe e α≥0 and n0is p ime o p. Since he p- h powe ope a ion pe mu es he ame cha ac e s o I0p ese ing hei o de , L⊗n0 ηmus be he i ial cha ac e . Now by [8, 1.14.2], Fis induced om a pα-dimensional ep esen a ion Go I0(n0), he unique open subg oup o I0o index n0. Then F ⊗ Lη= (IndI0 I0(n0)G)⊗ Lη∼ =IndI0 I0(n0)(G ⊗ ResI0 I0(n0)Lη) = IndI0 I0(n0)(G) = F since he es ic ion o Lη o I0(n0) is i ial.  We can now inish he p oo o heo em 2.2 o i educible ep esen a ions P oposi ion 3.7. Le F ∈ R0be i educible o slope a > 0. W i e a=c/d, whe e cand da e ela i ely p ime posi i e in ege s. Le Lηbe any ame cha ac e o I0 such ha L⊗d η=L⊗(c+d) χ. Then ρχ(F)∼ =F ⊗ Lη. P oo . Le nbe he dimension o F. By p oposi ions 3.4 and 3.5, he e exis s a ame cha ac e Lη0o I0such ha ρχ(F)∼ =F ⊗ Lη0, and L⊗n η0∼ =L⊗n(a+1) χ. Since he Swan conduc o na =nc/d o Fis an in ege , nmus be di isible by d. Then (L¯η0⊗ Lη)⊗n=L⊗n ¯η0⊗ L⊗d(n/d) η= L⊗n(a+1) ¯χ⊗ L⊗(c+d)n/d χ=L⊗n(a+1) ¯χ⊗ L⊗n(a+1) χ=1 so, by lemma 3.6, ρχ(F)∼ =F ⊗ Lη0∼ =(F ⊗ Lη0)⊗(L¯η0⊗ Lη) = F ⊗ Lη.  P oo o heo em 2.2. The unc o s Ra 0→ Ra 0gi en by F 7→ ρχ(F) and F 7→ F ⊗ Lηa e equi alences o ca ego ies, so hey p ese e di ec sums. I is enough hen o p o e he isomo phism o indecomposable ep esen a ions. 8 ANTONIO ROJAS-LE ´ ON So le F ∈ Ra 0be indecomposable o leng h m. Then by [10, Lemma 3.1.6, Lemma 3.1.7(3)] he e exis an i educible F0∈ Ra 0and a (necessa ily ame) inde- composable unipo en Um∈ R0o dimension msuch ha F=F0⊗Um. Since Fis a succesi e ex ension o mcopies o F0, by exac ness ρχ(F) is a succesi e ex ension o mcopies o ρχ(F0)∼ =F0⊗ Lη, which is i educible. By [10, Lemma 3.1.7(2)], he e is a unipo en U ∈ R0o dimension msuch ha ρχ(F)∼ =F0⊗ Lη⊗ U. Since ρχis an equi alence o ca ego ies, ρχ(F) mus be indecomposable, so U i sel mus be indecomposable. The e o e U∼ =Umand ρχ(F)∼ =F0⊗ Lη⊗ Um∼ =F ⊗ Lη.  4. Some a ian s We will conside now ep esen a ions o he ine ia g oup I∞a in ini y. Fo any F ∈ R∞o slope >1, we can ake i s local Fou ie ans o m FTψ (∞,∞)F, which is again in he same ca ego y. In [10, 3.4.4], N. Ka z asks abou a simple o mula o ρ0 χ(F) := FTψ,−1 (∞,∞)(L¯χ⊗FTψ (∞,∞)F), which is an au o-equi alence o he ca ego y o con inuous `-adic ep esen a ions o R∞wi h slopes >1. I can be in e p e ed as he wild pa o he monod omy a in ini y o he (addi i e) con olu ion F ∗ Lχ[10, 3.4.6], whe e Fis any ex ension o he ep esen a ion F o a smoo h shea on Gm,¯ k amely ami ied a 0. In his sec ion we will p o e Theo em 4.1. Le F ∈ R∞be o ally wild wi h a single slope a > 1. W i e a=c/d, whe e cand da e ela i ely p ime posi i e in ege s. Le Lηbe any ame cha ac e o I∞such ha L⊗d η=L⊗(c−d) ¯χ. Then ρ0 χ(F)∼ =F ⊗ Lη. In o he wo ds, we ha e he o mula (5) ρ0 χ(F)∼ =F ⊗ L⊗(a−1) ¯χ whe e L⊗(a−1) ¯χs ands o “any cha ac e ha can easonably be called L⊗(a−1) ¯χ”. The p oo is e y simila o he one o ρχ. Since e e y ep esen a ion in R∞ is a di ec sum o ep esen a ions wi h single slopes, we can assume ha Fhas a single slope a. Lemma 4.2. Le F,G ∈ R∞be o ally wild, wi h Fha ing all slopes >1. Then Swan(ρ0 χ(F)⊗ G) = Swan(F ⊗ G). P oo . We can assume ha Fhas a single slope a > 1. Ex end Fand G o smoo h shea es on Gm,¯ k, amely ami ied a 0, which we will also deno e by Fand G(as well as hei ex ensions by ze o o A1 ¯ k). Le nbe he ank o F, and deno e by F(0) i s local monod omy a 0, which is a ame ep esen a ion o I0. Since all slopes o Fa in ini y a e >1, i is a Fou ie shea [8, Lemma 8.3.1], so i s Fou ie ans o m is a single shea ha we will deno e by FTψF. By Ogg-Sha a e ic, FTψFis smoo h on Gm,¯ ko ank na. By Laumon’s local Fou ie ans o m heo y [9, Rema k 9], i has a single posi i e slope a a−1a in ini y wi h mul iplici y n(a−1) and ame pa isomo phic o d F(0), KATZ-RADON TRANSFORM OF `-ADIC REPRESENTATIONS 9 and i is un ami ied a 0. Then FTψF ⊗ L¯χalso has a single slope a a−1a in ini y wi h mul iplici y n(a−1), ame pa isomo phic o L¯χ⊗d F(0), and i s monod omy a 0 is a di ec sum o na copies o L¯χ. I s in e se Fou ie ans o m, by Ogg-Sha a e ic, is smoo h o ank n(a−1) a a−1+ n=n(a+ 1) on Gm,¯ k, and by local Fou ie ans o m i s monod omy a in ini y is he di ec sum o ρ0 χ(F) and na = Swan(F) copies o Lχ. A 0 is has i ial pa o ank na, whi h quo ien isomo phic o Lχ⊗ F(0). So Swan∞((F ∗ Lχ)⊗ G) = Swan(ρ0 χ(F)⊗ G) + Swan(F)Swan(G). We conclude exac ly as in lemma 3.3.  Using lemma 3.1 as in p oposi ion 3.4 we deduce P oposi ion 4.3. Le F ∈ R∞be i educible wi h slope >1. Then he e exis s a ame cha ac e Lηo I∞such ha ρ0 χ(F)∼ =F ⊗ Lη. P oposi ion 4.4. Le F ∈ R∞be i educible o dimension nand slope a > 1, and le Lηbe a ame cha ac e o I∞such ha ρ0 χ(F)∼ =F ⊗Lη. Then L⊗n η∼ =L⊗n(a−1) ¯χ. P oo . Ex end F o a smoo h `-adic shea on Gm,¯ k, amely ami ied a 0, also deno ed by F, and le Falso deno e i s ex ension by ze o o A1 ¯ k. By he p oo o lemma 4.2, he shea F ∗ Lχis smoo h on Gm,¯ k, i s monod omy a in ini y is he di ec sum o ρ0 χ(F)∼ =F ⊗ Lηand na copies o Lχ, and i s monod omy a 0 has i ial pa o dimension na wi h quo ien isomo phic o Lχ⊗ F(0). I s de e minan is hen a smoo h shea o ank 1 on Gm,¯ k, whose monod omy a ∞is de (F)⊗ L⊗n η⊗ L⊗na χ, and whose monod omy a 0 is de (F(0))⊗ L⊗n χ. We conclude, as in p oposi ion 3.5, ha de (F ∗ Lχ)∼ =de (F)⊗ L⊗n η⊗ L⊗na χ as shea es on Gm,¯ k. Compa ing hei monod omies a 0 gi es he desi ed isomo - phism.  The emainde o he p oo o heo em 4.1 is iden ical o he one o ρχ. We ha e a hi d a ian , o ep esen a ions F ∈ R∞wi h slopes <1: ρ00 χ(F) := FTψ,−1 (∞,0)(L¯χ⊗FTψ (∞,0)(F)), which is again an au o-equi alence o he ca ego y o con inuous `-adic ep esen a- ions o R∞wi h slopes <1. As in he ρχcase we ha e ρ00 χ(F)∼ =F ⊗ Lχ o F ame. The co esponding o mula o wild Fis Theo em 4.5. Le F ∈ R∞be o ally wild wi h a single slope a < 1. W i e a=c/d, whe e cand da e ela i ely p ime posi i e in ege s. Le Lηbe any ame cha ac e o I∞such ha L⊗d η=L⊗(d−c) χ. Then ρ00 χ(F)∼ =F ⊗ Lη. P oo . Le G:= FTψ (∞,0)(F)∈ R0, which has slope a 1−a=c d−c[9, Theo em 13]. The s a emen is hen equi alen o FTψ,−1 (∞,0)(L¯χ⊗ G)∼ =Lη⊗FTψ,−1 (∞,0)(G) o FTψ (∞,0)(Lη⊗FTψ,−1 (∞,0)(G)) ∼ =G ⊗ L¯χ.