Katz-Radon transform of l-adic representations

KATZ-RADON TRANSFORM OF `-ADIC REPRESENTATIONS ANTONIO ROJAS-LE ´ ON Abstract. We prove a simple explicit formula for the local Katz-Radon transform of an `-adic representation of the Galois group of the fraction field of a strictly henselian discrete valuation ring with positive residual characteristic, which can be defined as the local additive convolution with a fixed tame character. The formula is similar to one proved by D. Arinkin in the D-module setting, and answers a question posed by N. Katz. 1. Introduction In [10, 3.4.1], N. Katz defines some functors on the category of continuous `-adic representations of the inertia groups I0and I∞of the projective line over ¯ kat 0 and infinity, where ¯ kis the algebraic closure of a finite field of characteristic p and `is a prime different from p. These functors arise during his study of middle convolution of sheaves on the affine line and, roughly speaking, correspond to locally convolving a representation with a fixed tame character Lχof I0or I∞. They are defined using G. Laumon’s local Fourier transform functors, and in fact correspond to taking the tensor product with the conjugate tame character L¯χon the other side of the equivalence of categories given by these functors. Katz asks [10, 3.4.1] whether there is a simple expression for the functors defined in this way. Recently, D. Arinkin [1] has studied the analog of Katz’s functor in D-module theory: if Kis a field of characteristic 0, K((x)) is the field of Laurent series over Kand Dxthe ring of differential operators with coefficients in K((x)), the local Katz-Radon transform for a given λ∈K−Zis an equivalence of categories ρλ:Dx-mod→ Dx-mod, originally defined in [3]. Arinkin proves the simple formula [1, Theorem C] ρλ(F)∼ =F ⊗ Kλ(a+1) for any F ∈ Dx-mod with a single slope a, where Kµis the Kummer Dx-module of rank 1 generated by e, on which the derivative acts by d dxe=µ xe. In this article we will prove a similar formula in the `-adic case. More precisely, for a fixed tame `-adic character Lχand an `-adic representation Fof I0, let ρχ(F) := FTψ,−1 (0,∞)(L¯χ⊗FTψ (0,∞)F) where FTψ (0,∞)denotes Laumon’s local Fourier transform functor. If Fhas a single slope a=c/d (with c, d relatively prime positive integers), we will prove that there is an isomorphism of I0-representations ρχ(F)∼ =F ⊗ L⊗(a+1) χ Mathematics Subject Classification: 14F20,11F85,11S99 Partially supported by P08-FQM-03894 (Junta de Andaluc´ıa), MTM2010-19298 and FEDER. 1 2 ANTONIO ROJAS-LE ´ ON where L⊗(a+1) χis any d-th root of the character L⊗(c+d) χ. For a large class of representations Fof I0(in particular for many of those who appear in applications), the isomorphism can be proven via the explicit formulas for the local Fourier transforms given by L. Fu [5] and A. Abbes and T. Saito [2]. In this article we take a different approach that works for any F, and is independent of any explicit expression for the local Fourier transforms. 2. The Katz-Radon transform Fix a finite field kof characteristic p > 0 and an algebraic closure ¯ k. Let P1 ¯ k be the projective line over ¯ kand, for every t∈P1(¯ k) = ¯ k∪ {∞}, denote by Itits inertia group at t: for t6=∞, if x−tdenotes a local coordinate at t, it is the Galois group of the fraction field of the henselization of the local ring ¯ k[x](x−t). We have an exact sequence [8, 1.0] 0→Pt→It→Y `6=p Z`(1) →0 for every t∈P1(¯ k), where Ptis the only p-Sylow subgroup of It. Moreover, there is a canonical filtration of Itby the higher ramification groups I(r) t⊇I(s) tfor 0 ≤r < s ∈R which are normal in It. Fix a prime `6=p, and denote by Rtthe abelian category of continuous `-adic representations of It(i.e. continuous representations F:It→GLn(¯ Q`), whose image is in GLn(Eλ) for some finite extension Eλof Q`). For every irreducible F ∈ Rt, the slope of Fis inf{r≥0|F|I(r) t is trivial}. It is a non-negative rational number. In general, the slopes of Fare the slopes of the irreducible components of F. For every Fthere is a canonical direct sum decomposition [8, Lemma 1.8] (1) F∼ =M r≥0 Fr with Frhaving a single slope r. The slope 0 (tame) part will be denoted by Ft.F is said to be tame (respectively totally wild) if F=Ft(resp. Ft= 0). For every r≥0 let Rr tdenote the full subcategory of Rtconsisting of representations with a single slope r. We have a decomposition Rt=M r≥0 Rr t in the sense that every F ∈ Rthas a decomposition (1) and HomRt(F,G) = 0 if F ∈ Rr t,G ∈ Rs tand r6=s[8, Proposition 1.1]. Let k0⊆¯ kbe a finite extension of k, and χ:k0∗ →¯ Q∗ `a multiplicative character. By [4, 1.4-1.8] there is an associated smooth Kummer sheaf Lχon Gm,¯ k, which is a tame character of I0(and of I∞) of the same order as χ. If k0⊆k00 is another extension, the sheaves defined by χand χ◦Nmk00 /k0:k00∗ →¯ Q∗ `are isomorphic. Moreover, every tame character of I0(and of I∞) can be obtained in this way. Whenever we speak about a tame character of I0, we will implicitly assume that we have made a choice of such a finite extension of kand of a character. KATZ-RADON TRANSFORM OF `-ADIC REPRESENTATIONS 3 Fix a non-trivial additive character ψ:k→¯ Q∗ `. The local Fourier transform functors, defined by G. Laumon in [11], give equivalences of categories FTψ (0,∞):R0→ R<1 ∞, FTψ (∞,∞):R>1 ∞→ R>1 ∞ and FTψ (∞,0) :R<1 ∞→ R0 (where R<1 ∞=Lr<1Rr ∞and R>1 ∞=Lr>1Rr ∞) that describe the relationship between the local monodromies of an `-adic sheaf on A1 ¯ kand its Fourier transform with respect to ψ. The Katz-Radon transform is defined in terms of them. Definition 2.1. Fix a tame character Lχof I0. The (local) Katz-Radon transform (with respect to Lχ) is the functor ρχ:R0→ R0given by ρχ(F) = FTψ,−1 (0,∞)(FTψ (0,∞)Lχ⊗FTψ (0,∞)F) = FTψ,−1 (0,∞)(L¯χ⊗FTψ (0,∞)F). The Katz-Radon transform is an auto-equivalence of the category R0(since it is a composition of three equivalences of categories). It preserves dimensions and slopes, and for tame Fit is given by ρχ(F) = F ⊗ Lχ[10, 3.4.1]. For totally wild F, it can be interpreted as the “local additive convolution” of Fand Lχ[10, 3.4.3]: if we extend Fto a smooth sheaf on Gm,¯ k, tamely ramified at infinity, then ρχ(F) is the wild part of the local monodromy at 0 of F ∗ Lχ, where F ∗ Lχ= R1σ!(FLχ) and σ:A2 ¯ k→A1 ¯ kdenotes the addition map (in [10], the “middle convolution” is used instead, but that one differs from the one used here only by Artin-Shreier components, which are smooth at 0 and therefore do not affect the local monodromy). Notice that, in particular, ρχis independent of the choice of the additive character ψ. More intrinsically, it can be described in terms of vanishing cycles functors [11, 2.7.2]: If X=A2 (0,0) (respectively S=A1 (0)) denotes the henselization of A2 ¯ kat (0,0) (resp. the henselization of A1 ¯ kat 0) then ρχ(F)∼ =R1Φ(σ, FLχ)(0,0), where RΦ(σ, FLχ) is the vanishing cycles complex for the addition map σ:X→S with respect to the sheaf FLχon X. Similarly, it also has an interpretation as a “local multiplicative convolution” [12, Corollary 5.6]: If X=G2 m,(1,1) (respectively S=Gm,(1)) denotes the henselization of Gm,¯ kat (1,1) (resp. the henselization of Gm,¯ kat 1) then ρχ(F)∼ =R1Φ(µ, F Lχ)(1,1), where RΦ(µ, FLχ) is the vanishing cycles complex for the multiplication map µ:X→Swith respect to the sheaf FLχon X, and Fand Lχare viewed as representations of I1via the isomorphism I0∼ =I1that maps the uniformizer x at 0 to the uniformizer x−1 at 1. The main result of this article is the following simple expression for ρχ: Theorem 2.2. Let F ∈ R0be totally wild with a single slope a > 0. Write a=c/d, where cand dare relatively prime positive integers. Let Lηbe any tame character of I0such that L⊗d η=L⊗(c+d) χ. Then ρχ(F)∼ =F ⊗ Lη. 4 ANTONIO ROJAS-LE ´ ON In other words, we have the formula (2) ρχ(F)∼ =F ⊗ L⊗(a+1) χ where L⊗(a+1) χstands for “any character that can reasonably be called L⊗(a+1) χ”. By the decomposition R0=Lr≥0Rr 0, this determines ρχ(F) for any F ∈ R0, thus answering the question posed by N. Katz in [10, 3.4.1]. A question that remains open is the following: in the article we prove that ρχ(F)∼ =F ⊗ Lη, independently for any Fwith slope a. So the functors Ra 0→ Ra 0 given by ρχand (−)⊗ Lηmap any Fto isomorphic objects. Is there an actual isomorphism of functors between them? In the affirmative case, is there a simple way to construct it? 3. Proof of the main theorem In this section we will prove theorem 2.2. We will start with the case where F ∈ R0is irreducible. Lemma 3.1. Let F ∈ R0. Then Ft6= 0 if and only if there exists  > 0such that for every G ∈ R0with a single slope b∈(0, )we have Swan(F ⊗ G)>Swan(F) dim(G). Proof. Suppose that Ft6= 0, and let a0= 0 < a1<· · · < arbe the slopes of F, with multiplicities n0, n1, . . . , nr. Then Swan(F) = Pniai. Let =a1. Then for every G ∈ R0with a single slope b∈(0, ) the tensor product F ⊗ G has slopes b<a1<· · · < arwith multiplicities n0m, n1m, . . . , nrmwhere m= dim(G) by [8, Lemma 1.3]. Therefore Swan(F ⊗ G) = n0mb + r X i=1 nimai> r X i=1 nimai= Swan(F) dim(G). Conversely, suppose that Ft= 0, and let a1<· · · < arbe the slopes of F. Then for every G ∈ R0with a single slope b∈(0, a1) the tensor product F ⊗ G has the same slopes as Fby [8, Lemma 1.3], and in particular Swan(F ⊗ G) = Swan(F) dim(G). This proves the lemma, since for every  > 0 there exist representations in R0with slope b∈(0, ) (for instance, one may take [n]∗H, where H ∈ R0has slope a > 0 and nis a prime to pinteger greater than a/ [8, 1.13.2]).  For any two objects K, L ∈ Db c(A1 ¯ k,¯ Q`), we will denote by K∗L∈ Db c(A1 ¯ k,¯ Q`) their additive convolution: K∗L= Rσ!(KL) where σ:A2 ¯ k→A1 ¯ kis the addition map. Lemma 3.2. Let K, L, M ∈ Db c(A1 ¯ k,¯ Q`). Then RΓc(A1 ¯ k,(K∗L)⊗M)∼ =RΓc(A1 ¯ k, K ⊗((τ∗ −1L)∗M)) where τ−1:A1 ¯ k→A1 ¯ kis the additive inversion. Proof. We have 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 the projection formula. If π1, π2:A2 ¯ k→A1 ¯ kare the projections then RΓc(A2 ¯ k,(KL)⊗σ∗M) = RΓc(A2 ¯ k, π∗ 1K⊗π∗ 2L⊗σ∗M). Consider the automorphism φ:A2 ¯ k→A2 ¯ kgiven by (x, y)7→ (x+y, −y). Then σ=π1◦φ,π1=σ◦φand τ−1◦π2=π2◦φ. It follows that 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)).  If Fis a smooth ¯ Q`-sheaf on Gm,¯ kwhich is totally wild at 0, then for every t∈¯ kthe sheaf F ⊗Lχ(t−x)(extended by zero to A1 ¯ k) is totally wild at 0 and has no punctual sections (where Lχ(t−x)is the pull-back of the Kummer sheaf Lχunder the map x7→ t−x), so its only non-zero cohomology group with compact support is H1 c. We conclude that the only non-zero cohomology sheaf of F[0] ∗ Lχ[0] ∈ Db c(A1 ¯ k,¯ Q`) is H1= R1σ!(F ⊗ Lχ). We will denote this sheaf by F ∗ Lχ. Lemma 3.3. Let F,G ∈ R0. Then Swan(ρχ(F)⊗ G) = Swan(F ⊗ G). Proof. By additivity of the Swan conductor, we may assume that Fis irreducible, and in particular that it has a single slope a≥0. If a= 0 then ρχ(F)∼ =F ⊗ Lχ, so the equality is clear. Suppose that a > 0. By [7, Theorem 1.5.6], Fand Gcan be extended to smooth sheaves on Gm,¯ k, tamely ramified at infinity, which we will also denote by Fand G. Let Fand Gbe also their extensions by zero to A1 ¯ k. Using the compatibility between Fourier transform with respect to ψand convolution [11, Proposition 1.2.2.7], we have F ∗ Lχ= FT ¯ ψ(FTψF ⊗ FTψLχ) = FT ¯ ψ(FTψF ⊗ L¯χ), where FTψFdenotes the “naive” Fourier transform in the sense of [8, 8.2], that is, the (−1)-th cohomology sheaf of the Fourier transform of F[1] ∈ Db c(A1 ¯ k,¯ Q`) (which is its only non-zero cohomology sheaf, since Fis totally wild at zero and therefore it is Fourier [8, Lemma 8.3.1]). Let nbe the rank of F, and denote by F(∞)∈ R∞its local monodromy at infinity, which is a tame representation of I∞. By Ogg-Shafarevic [6, Expos´e X, Corollaire 7.12], FTψFis smooth on Gm,¯ kof rank na+n=n(a+1). By Laumon’s local Fourier transform theory [9, Theorem 13], FTψFhas a single slope a a+1 at infinity, with multiplicity n(a+ 1), and its monodromy at 0 has a trivial part of dimension na and its quotient is the dual [ F(∞)of F(∞). Then FTψF ⊗L¯χalso has a single slope a a+1 at infinity with multiplicity n(a+ 1), and its monodromy Mat 0 sits in an exact sequence (3) 0 → L⊕na ¯χ→M→ [ F(∞)⊗ L¯χ→0. Its inverse Fourier transform, by Ogg-Shafarevic, is smooth of rank n(a+1) on Gm,¯ k, and by local Fourier transform its wild part at 0 has slope awith multiplicity n. 6 ANTONIO ROJAS-LE ´ ON In fact, this wild part is simply ρχ(F) by the additive convolution interpretation of ρχ. Its monodromy at infinity sits in an exact sequence (4) 0 → L⊕na χ→(F ∗ Lχ)(∞)→ F(∞)⊗ Lχ→0 obtained from (3) by local Fourier transform. So F ∗ Lχhas rank n(a+ 1) on Gm,¯ k, and its monodromy at 0 is the direct sum of ρχ(F) and a constant part of dimension na = Swan(F). So Swan0((F ∗ Lχ)⊗ G) = Swan(ρχ(F)⊗ G) + Swan(F)Swan(G). In particular, by Ogg-Shafarevic, the Euler characteristic of the sheaf (F ∗ Lχ)⊗ G (extended by zero to A1 ¯ k) is −Swan(ρχ(F)⊗ G)−Swan(F)Swan(G). Using lemma 3.2, proper base change, and the fact that χ(Gm,¯ k, K ⊗ Lχ) = χ(Gm,¯ k, K) for any object K∈ Db c(Gm,¯ k,¯ Q`), we get 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]) −rank0(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) where rank0of a derived category object denotes the alternating sum of the ranks at 0 of its cohomology sheaves, so Swan(ρχ(F)⊗ G) = Swan(F ⊗ G).  Proposition 3.4. Let F ∈ R0be totally wild and irreducible. Then there exists a tame character Lηof I0such that ρχ(F)∼ =F ⊗ Lη. Proof. Let b Fbe the dual representation. We claim that the tame part of ρχ(F)⊗b F is non-zero. By lemma 3.1, it suffices to show that there is an  > 0 such that, for any G ∈ R0with slope b∈(0, ), Swan(ρχ(F)⊗b F ⊗G)>Swan(ρχ(F)⊗b F) dim(G). But by lemma 3.3, we have 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 the dual of F, the tensor product F ⊗ b Fhas a trivial quotient and, in particular, has non-trivial tame part. By lemma 3.1, there exists  > 0 such that, for any G ∈ R0with slope b∈(0, ), Swan(F ⊗ b F ⊗ G)>Swan(F ⊗ b F) dim(G). Since the tame part of ρχ(F)⊗b Fis non-zero and it is a direct summand, it contains a tame character Lηof I0as a subrepresentation. Then ρχ(F)⊗b F ⊗ L¯η=ρχ(F)⊗\ F ⊗ Lη= Hom(F ⊗ Lη, ρχ(F)) contains a trivial subrepresentation, so HomI0(F ⊗ Lη, ρχ(F)) 6= 0. Since both ρχ(F) and F ⊗Lηare irreducible, any non-zero I0-equivariant map F ⊗Lη→ρχ(F) must be an isomorphism.  Proposition 3.5. Let F ∈ R0be totally wild and irreducible of dimension nand slope a, and let Lηbe a tame character of I0such that ρχ(F)∼ =F ⊗ Lη. Then L⊗n η∼ =L⊗n(a+1) χ. KATZ-RADON TRANSFORM OF `-ADIC REPRESENTATIONS 7 Proof. Extend Fto a smooth `-adic sheaf on Gm,¯ k, tamely ramified at infinity, also denoted by F. Let Falso denote its extension by zero to A1 ¯ k. By the proof of lemma 3.3, the sheaf F ∗ Lχis smooth on Gm,¯ k, its monodromy at 0 is the direct sum of ρχ(F)∼ =F ⊗ Lηand a trivial part of dimension na, and its monodromy at infinity sits in the exact sequence (4). Its determinant is then a smooth sheaf of rank 1 on Gm,¯ k, whose monodromy at 0 is det(F)⊗ L⊗n η, and whose monodromy at ∞is det(F(∞))⊗ L⊗n(a+1) χ. Then \ det(F)⊗ L⊗n ¯η⊗det(F ∗ Lχ) is a rank 1 smooth sheaf on Gm,¯ k, with trivial monodromy at 0 and tamely ramified at infinity. Since the tame fundamental group of A1 ¯ kis trivial, we conclude that det(F ∗ Lχ)∼ =det(F)⊗ L⊗n η as sheaves on Gm,¯ k. Comparing their monodromies at infinity gives the desired isomorphism.  It remains to show that any such Lηworks. Lemma 3.6. Let F ∈ R0be irreducible of dimension n, and let Lηbe a tame character of I0such that L⊗n ηis trivial. Then F ⊗ Lη∼ =F. Proof. Write n=n0pα, where α≥0 and n0is prime to p. Since the p-th power operation permutes the tame characters of I0preserving their order, L⊗n0 ηmust be the trivial character. Now by [8, 1.14.2], Fis induced from a pα-dimensional representation Gof I0(n0), the unique open subgroup of I0of index n0. Then F ⊗ Lη= (IndI0 I0(n0)G)⊗ Lη∼ =IndI0 I0(n0)(G ⊗ ResI0 I0(n0)Lη) = IndI0 I0(n0)(G) = F since the restriction of Lηto I0(n0) is trivial.  We can now finish the proof of theorem 2.2 for irreducible representations Proposition 3.7. Let F ∈ R0be irreducible of slope a > 0. Write a=c/d, where cand dare relatively prime positive integers. Let Lηbe any tame character of I0 such that L⊗d η=L⊗(c+d) χ. Then ρχ(F)∼ =F ⊗ Lη. Proof. Let nbe the dimension of F. By propositions 3.4 and 3.5, there exists a tame character Lη0of I0such that ρχ(F)∼ =F ⊗ Lη0, and L⊗n η0∼ =L⊗n(a+1) χ. Since the Swan conductor na =nc/d of Fis an integer, nmust be divisible 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.  Proof of theorem 2.2. The functors Ra 0→ Ra 0given by F 7→ ρχ(F) and F 7→ F ⊗ Lηare equivalences of categories, so they preserve direct sums. It is enough then to prove the isomorphism for indecomposable representations. 8 ANTONIO ROJAS-LE ´ ON So let F ∈ Ra 0be indecomposable of length m. Then by [10, Lemma 3.1.6, Lemma 3.1.7(3)] there exist an irreducible F0∈ Ra 0and a (necessarily tame) indecomposable unipotent Um∈ R0of dimension msuch that F=F0⊗Um. Since Fis a succesive extension of mcopies of F0, by exactness ρχ(F) is a succesive extension of mcopies of ρχ(F0)∼ =F0⊗ Lη, which is irreducible. By [10, Lemma 3.1.7(2)], there is a unipotent U ∈ R0of dimension msuch that ρχ(F)∼ =F0⊗ Lη⊗ U. Since ρχis an equivalence of categories, ρχ(F) must be indecomposable, so U itself must be indecomposable. Therefore U∼ =Umand ρχ(F)∼ =F0⊗ Lη⊗ Um∼ =F ⊗ Lη.  4. Some variants We will consider now representations of the inertia group I∞at infinity. For any F ∈ R∞of slope >1, we can take its local Fourier transform FTψ (∞,∞)F, which is again in the same category. In [10, 3.4.4], N. Katz asks about a simple formula for ρ0 χ(F) := FTψ,−1 (∞,∞)(L¯χ⊗FTψ (∞,∞)F), which is an auto-equivalence of the category of continuous `-adic representations of R∞with slopes >1. It can be interpreted as the wild part of the monodromy at infinity of the (additive) convolution F ∗ Lχ[10, 3.4.6], where Fis any extension of the representation Fto a smooth sheaf on Gm,¯ ktamely ramified at 0. In this section we will prove Theorem 4.1. Let F ∈ R∞be totally wild with a single slope a > 1. Write a=c/d, where cand dare relatively prime positive integers. Let Lηbe any tame character of I∞such that L⊗d η=L⊗(c−d) ¯χ. Then ρ0 χ(F)∼ =F ⊗ Lη. In other words, we have the formula (5) ρ0 χ(F)∼ =F ⊗ L⊗(a−1) ¯χ where L⊗(a−1) ¯χstands for “any character that can reasonably be called L⊗(a−1) ¯χ”. The proof is very similar to the one for ρχ. Since every representation in R∞ is a direct sum of representations with single slopes, we can assume that Fhas a single slope a. Lemma 4.2. Let F,G ∈ R∞be totally wild, with Fhaving all slopes >1. Then Swan(ρ0 χ(F)⊗ G) = Swan(F ⊗ G). Proof. We can assume that Fhas a single slope a > 1. Extend Fand Gto smooth sheaves on Gm,¯ k, tamely ramified at 0, which we will also denote by Fand G(as well as their extensions by zero to A1 ¯ k). Let nbe the rank of F, and denote by F(0) its local monodromy at 0, which is a tame representation of I0. Since all slopes of Fat infinity are >1, it is a Fourier sheaf [8, Lemma 8.3.1], so its Fourier transform is a single sheaf that we will denote by FTψF. By Ogg-Shafarevic, FTψFis smooth on Gm,¯ kof rank na. By Laumon’s local Fourier transform theory [9, Remark 9], it has a single positive slope a a−1at infinity with multiplicity n(a−1) and tame part isomorphic to d F(0), KATZ-RADON TRANSFORM OF `-ADIC REPRESENTATIONS 9 and it is unramified at 0. Then FTψF ⊗ L¯χalso has a single slope a a−1at infinity with multiplicity n(a−1), tame part isomorphic to L¯χ⊗d F(0), and its monodromy at 0 is a direct sum of na copies of L¯χ. Its inverse Fourier transform, by Ogg-Shafarevic, is smooth of rank n(a−1) a a−1+ n=n(a+ 1) on Gm,¯ k, and by local Fourier transform its monodromy at infinity is the direct sum of ρ0 χ(F) and na = Swan(F) copies of Lχ. At 0 is has trivial part of rank na, whith quotient isomorphic to Lχ⊗ F(0). So Swan∞((F ∗ Lχ)⊗ G) = Swan(ρ0 χ(F)⊗ G) + Swan(F)Swan(G). We conclude exactly as in lemma 3.3.  Using lemma 3.1 as in proposition 3.4 we deduce Proposition 4.3. Let F ∈ R∞be irreducible with slope >1. Then there exists a tame character Lηof I∞such that ρ0 χ(F)∼ =F ⊗ Lη. Proposition 4.4. Let F ∈ R∞be irreducible of dimension nand slope a > 1, and let Lηbe a tame character of I∞such that ρ0 χ(F)∼ =F ⊗Lη. Then L⊗n η∼ =L⊗n(a−1) ¯χ. Proof. Extend Fto a smooth `-adic sheaf on Gm,¯ k, tamely ramified at 0, also denoted by F, and let Falso denote its extension by zero to A1 ¯ k. By the proof of lemma 4.2, the sheaf F ∗ Lχis smooth on Gm,¯ k, its monodromy at infinity is the direct sum of ρ0 χ(F)∼ =F ⊗ Lηand na copies of Lχ, and its monodromy at 0 has trivial part of dimension na with quotient isomorphic to Lχ⊗ F(0). Its determinant is then a smooth sheaf of rank 1 on Gm,¯ k, whose monodromy at ∞is det(F)⊗ L⊗n η⊗ L⊗na χ, and whose monodromy at 0 is det(F(0))⊗ L⊗n χ. We conclude, as in proposition 3.5, that det(F ∗ Lχ)∼ =det(F)⊗ L⊗n η⊗ L⊗na χ as sheaves on Gm,¯ k. Comparing their monodromies at 0 gives the desired isomorphism.  The remainder of the proof of theorem 4.1 is identical to the one for ρχ. We have a third variant, for representations F ∈ R∞with slopes <1: ρ00 χ(F) := FTψ,−1 (∞,0)(L¯χ⊗FTψ (∞,0)(F)), which is again an auto-equivalence of the category of continuous `-adic representations of R∞with slopes <1. As in the ρχcase we have ρ00 χ(F)∼ =F ⊗ Lχfor F tame. The corresponding formula for wild Fis Theorem 4.5. Let F ∈ R∞be totally wild with a single slope a < 1. Write a=c/d, where cand dare relatively prime positive integers. Let Lηbe any tame character of I∞such that L⊗d η=L⊗(d−c) χ. Then ρ00 χ(F)∼ =F ⊗ Lη. Proof. Let G:= FTψ (∞,0)(F)∈ R0, which has slope a 1−a=c d−c[9, Theorem 13]. The statement is then equivalent to FTψ,−1 (∞,0)(L¯χ⊗ G)∼ =Lη⊗FTψ,−1 (∞,0)(G) or FTψ (∞,0)(Lη⊗FTψ,−1 (∞,0)(G)) ∼ =G ⊗ L¯χ.