9 Artin representations

Transcription

1 9 Artin representations Let K be a global field. We have enough for G ab K. Now we fix a separable closure Ksep and G K := Gal(K sep /K), which can have many nonabelian simple quotients. An Artin representation (ρ, V ) is a continuous homomorphism ρ : G K GL(V ) for some finite dimensional C-vector space V. Here GL(V ) is the space of automorphisms of V equipped with the natural topology from that of C. We say (ρ, V ) and (ρ, V ) are equivalent if there is an isomorphisms f : V V such that ρ = f ρ f 1. We identify equivalent Artin representations. It is common to abbreviate (ρ, V ) to ρ when there is no ambiguity. For any decomposition of vector space V = V 1 V 2, we have an inclusion of groups ι : GL(V 1 ) GL(V 2 ) GL(V ) where GL(V 1 ) acts trivially on V 2 and vise versa. For ρ i : G K GL(V i ), i = 1, 2, we say ρ = ρ 1 ρ 2 if ρ = ι (ρ 1, ρ 2 ) and say ρ is the direct sum of the subrepresentations ρ 1 and ρ 2. If this happens for some V 1, V 2 non-trivial we say ρ is reducible, and irreducible otherwise. Representation theory of the finite group G K / ker(ρ) gives Lemma 9.1. Every Artin representation is a unique direct sum (up to permutation) of irreducible subrepresentations. We have also seen Lemma 9.2. For any Artin representation ρ, Im(ρ) is finite. The kernel of ρ is a finite index open normal subgroup, and thus ker ρ = for some finite Galois extension L/K. Recall also that for any place v for K, we can fixed an embedding of separable closures K sep Kv sep, which induces ι v : G Kv G K. Different choices simply conjugate ι v by elements in G K. Definition 9.3. ρ is said to be unramified (resp. tamely ramified) at a non-archimedean place v iff the following equivalent condition holds: (i) L/K is unramified (resp. tamely ramified) at v. (ii) The image of the inertia I v (resp. wild inertia P v ) of G Kv is contained in ker ρ. It is obvious that any Artin representation is unramified almost everywhere. When ρ is unramified at v, the composition G Kv G K ρ GL(V ) factors through GKv /I v = Frob v. In general, let V Iv V be the subspace fixed by ρ(i v ), then Frob v acts on V Iv. Definition 9.4. Let (ρ, V ) be an Artin representation and v a non-archimedean place of K. We define the Artin local L-factor Recall Γ R (s) = π s/2 Γ(s/2). L v (s, ρ) := det(id V Iv q s v ρ(frob v ) V Iv ) 1. 1

2 Definition 9.5. Let (ρ, V ) be an Artin representation and v an archimedean place of K. We define the Artin local L-factor at v as follows: either K v = R and ρ = (triv) n 1 + (sgn) n 2, or K v = C and ρ = (triv) dim V is trivial. In the former case, we put L v (s, ρ) = Γ R (s) n 1 Γ R (s + 1) n 2. In the latter case put L v (s, ρ) = Γ R (s) dim V Γ R (s + 1) dim V. Remark 9.6. Despite that we chose a global language, one sees that all these definitions are purely local. For example, for F a local field and ρ : G F GL(V ) a continuous representation (i.e. homomorphism) we can define L F (s, ρ) likewise, so that L v (s, ρ) = L Kv (s, ρ GKv ). Definition 9.7. Let (ρ, V ) be an Artin representation. The Artin L-function is the Euler product L(s, ρ) = L v (s, ρ) v The Euler product converges again for Re(s) > 1. This is because as Im(ρ) is finite, Frob v is diagonalizable with eigenvalues being roots of unity. One sees directly that when dim V = 1, our Artin representations become Dirichlet characters, and our Artin L-functions agrees with the Dirichlet L-functions we had last time. In fact, they are compatible in even greater generality. Suppose G is any group and H G a (say finite index) subgroup, and ρ H : H GL(V ) a representation of H. One may think of V as a left C[H]-module, and form C[G] C[H] V as a left C[G]-module. This gives a representation ρ G : G GL(C[G] C[H] V ) and we write ind G H ρ H := ρ G. For any finite separable extension L/K and an Artin representation ρ L : GL(V ) for L, one checks that ind G K ρ L is an Artin representation for K. Proposition 9.8. Let L/K be a finite separable extension and ρ L be an Artin representation for L. Then L v (s, ind G K ρ L ) = w v L w (s, ρ L ) where w runs over places of L above v. Corollary 9.9. Let L/K be a finite separable extension and ρ L be an Artin representation for L. Then L(s, ind G K ρ L ) = L(s, ρ L ) where the first Artin L-function is of the Artin representation ind G K ρ L for K. To prove the proposition, we first establish its purely local part: 2

3 Lemma Let E/F be a finite separable extension of local fields and ρ E : GL(V ) a continuous representation. Then for L F (s, ind G F ρ E ) = L E (s, ρ E ). Proof. The case for F archimedean is obvious; we assume F non-archimedean. Let E be the maximal unramified subextension of E/F. Then we have from definition ind G F ρ E = ind G F G ind G E G E E ρ E, and the desired equality will follow from the two equalities L F (s, ind G F ρ E ) = L E (s, ind G E ρ E ) = L E (s, ρ E ). In other words, it suffices to deal with the case E = E and F = E. Let us write ρ F = ind G F ρ E. Also write I F the inertia in G F and likewise I E = I F the inertia in. First we consider the case E = F, so that E/F is totally ramified and equivalently G F = I F. In particular G F /I F = GE /I E and they share the same Frobenius σ. Let {θ 0 = 1, θ 1..., θ e 1 } be a set of representatives of G F / = I F / = IF /I E. Then e 1 V := C[G F ] C[GE ] V = θ i V and one has V I E V I F by v (v, θ 1 (v),...,, θ e 1 (v)). This is actually an isomorphism; every element in V fixed by I F, i.e. by I E and by θ 1,..., θ e 1 has to be of this form. Hence L F (s, ind G F ρ E ) = det(id V I F ρ F (σ) V I F ) 1 = det(id V I E ρ E (σ) V I E ) 1 = L E (s, ρ E ). Now suppose E = E, so that E/F is unramified and equivalently I F = I E. Let σ G F be any Frobenius. We have {1,..., σ f 1 } as a set of representatives of G F /, and σ f is a Frobenius in. We have f 1 f 1 V := C[G F ] C[GE ] V = σ i V, and V I F = σ i V I E. i=0 i=0 i=0 By the elementary properties of determinants this gives L F (s, ind G F ρ E ) = det(id V I F ρ F (σ) V I F ) 1 = det(id V I E ρ E (σ n ) V I E ) 1 = L E (s, ρ E ). Proof of Proposition 9.8. Now we are ready to prove the global result. The idea is that if {α i } is a set of representatives of G K /. Then we have V := C[G K ] C[GL ]V = i α iv. Now Frob v acts on V. The action is related to the action of G Kv on G K / We have a G Kv -equivariant (left action) bijection G K / = G Kv /w. w v Thus one has V = w v C[G K v ] C[GLw ] V as representations of G Kv, which is enough for defining L-factors. The desired identity thus follows from Lemma

4 Example Let L/K be any quadratic (separable) extension of global field. Let triv K (resp. triv L ) be the trivial representation of G K (resp. ) and χ L/K be the non-trivial character of Gal(L/K) pulled-back to G K. One has ind G K ind L = ind K χ L/K, and thus by Corollary 9.9 we have L(s, ind L ) = L(s, ind G K ind K ) = L(s, ind K )L(s, χ L/K ) which is well-known at least when K = Q. By the standard character theory of finite (or compact Hausdorff) groups, we know we can identify representations of Gal(L/K) (or G K ) as characters, i.e. certain functions on Gal(L/K) (or G K ), and one can talk about possibly negative integral linear combination of representations as virtual representations. Suppose now ρ as a representation of G = Gal(L/K) (or G = G K ) is an integral combination of certain ind G H χ for finite index open subgroups H G and 1-dimensional representations χ : H C. Then by Corollary 9.9 and our results for L-functions of Dirichlet characters, we see that L(s, ρ) is meromorphic for s C. This is indeed the case by a theorem of Brauer Theorem (Brauer) Let G be a finite group. Then any representation of G is an integral linear combination of inductions of 1-dimensional characters of subgroups H G. In fact, all H can be chosen among subgroups for which each is a direct product of a cyclic group and a p-group. Proof. We refer to the 7-page paper of Brauer and Tate, On the characters of finite groups, Ann. of Maths., 62, As taking dual of a representation is a conjugate-linear action on the space of characters, we have a functional equation relating L(s, ρ) and L(1 s, ρ ) where ρ is the dual representation. By Brauer s theorem we thus have Corollary Let ρ be any Artin representation. Then L(s, ρ) is meromorphic for s C, and there exists constants C(ρ), N(ρ) Q >0 as well as ɛ(ρ) C, ɛ(ρ) = 1 such that L(1 s, ρ ) = (C(ρ)N(ρ)) 1/2 ɛ(ρ)l(s, ρ). We would like to say a few words about C(ρ), N(ρ) and ɛ(ρ). Recall that when dim(ρ) = 1, we had N = v q nv v where the product runs over non-archimedean places v and n v is such that our chosen 1 additive character ψ K is trivial on ϖv nv O Kv but non-trivial on ϖv nv 1 O Kv. When K is a number field, N is the norm of the different to Q. When K contain F q as its constant field, we have N = q 2g 2. 1 When one scales the choice by some x K, the product formula ensures that N is scaled by x = 1, thus invariant. 4

5 Now we would like to put simply N(ρ) = N dim(ρ) ; in particular N(ρ) Z when K is a number field. It is then not an easy task to find a general formula for C(ρ). This had also been carried out by E. Artin and involves substantial interplay with ramification groups and class field theory. Recall that for a finite Galois extension of local fields E/F with G = Gal(E/F ), the lower ramification groups are defined by G i = {σ G σ(x) x ϖ i+1 E O E, x O E }, i 0 so that G 0 is the inertia and G 1 is the wild inertia. We define c E/F : G C as 0, g G 0 c E/F (g) = (i + 1) G G 0, g G i\g i+1 g G\{1} c E/F (g), g = 1 One directly computes that for any representation ρ of G with character χ ρ : G C we have (c E/F, χ ρ ) = 1 c E/F (g)χ ρ (g) = 1 χ ρ (1) χ ρ (σ). G G 0 g G i 0 σ G i Theorem (Artin) (c E/F, χ ρ ) Z 0. In other words, c is the character of some representation. Now let ρ be an Artin representation for our global field K which factor through Gal(L/K) for some L/K finite Galois. For each non-archimedean place v, take any w v of L, let G v = Gal(L w /K v ) Gal(L/K) and let χ ρv be the character of ρ Gv. Let the Artin conductor c(ρ) be c(ρ) := p (c Lw/Kv,χρv ) v. v and we also put C(ρ) := N(c(ρ)) = [O K : c(ρ)]. (When K is a global field, we should think of c(ρ) as a divisor and C(ρ) = q deg(c(ρ)).) In particular we see C(ρ) Z. Theorem (Artin) Our C(ρ) and N(ρ) are what is needed in Corollary There remains the question of expressing ɛ(ρ). It will be desirable to write ɛ(ρ) = v ɛ v(ρ, ψ). This turns out to be difficult: Theorem (Deligne) There exist unique constants ɛ F (ρ, ψ F ) C of absolute value 1 for any local field F, non-trivial additive character ψ F and virtual C-representation ρ of G F, such that ɛ F (ρ, ψ F ) is the same as what we had when ρ is a 1-dimensional (actual) represen- (i) tation. (ii) ɛ F (ρ 1 ρ 2, ψ F ) = ɛ F (ρ 1, ψ F )ɛ F (ρ 2, ψ F ). 5

6 (iii) Suppose E/F is a finite extension, ψ E = ψ F Tr E/F, and ρ E is any virtual representation of degree 0 of, we have ɛ E (ρ E, ψ E ) = ɛ F (ind G F ρ E, ψ F ). (iv) Let K be our global field with chosen ψ K : A K C. Write ψ Kv ɛ(ρ) = ɛ Kv (ρ GKv, ψ Kv ). v := ψ K Kv. Then Lastly, we mention the Artin Conjecture, which is one of the many motivations for Langlands program: Conjecture Let ρ be an irreducible non-trivial Artin representation. Then L(s, ρ) is entire. In the case when K is a global function field (say the function field of the algebraic curve X), the conjecture was proved by Weil in the case when K is a global function field. Nowadays this case is a standard application of Grothendieck-Lefschetz fixed point theorem, which allows us to express L(s, ρ) in terms of the l-adic cohomology of the local system on X associated to ρ. (In particular, the poles of L(s, ρ) comes from even degree l-adic cohomology in Hét even (X, ρ) = Hét 0 (X, ρ) H2 ét (X, ρ) = Hét 0 (X, ρ) H0 ét (X, ρ ) = 0, thus there are no poles.) 10 l-adic representations We fix a prime l. Artin representations are not enough; l-adic representations with infinite images naturally appear in geometry. Definition Let K be any field. An l-adic Galois representation (for K) is a continuous homomorphism ρ : G K GL(V ) where V is a finite-dimensional vector space over Q l. Example 1. There is a natural surjection G Q G ab Q = Ẑ. Explicitly, Q ab = Q(ζ ) is the extension of Q with all roots of unity. The Galois group G Q acts on the m-th roots of unity, giving a map G Q (Z/mZ). Taking the (inverse) limit of all such maps gives G Q Ẑ. One may also concentrate at a prime l by taking Ẑ Z l. This amounts to looking at the action of G Q on all l n -th roots of unity. We call the Galois representation χ cyc : G Q Z l the (l-adic) cyclotomic character. We have a geometric interpretation of the cyclotomic character: Let G m be the variety for which G m (k) = {(a, b) k 2 ab = 1} for any field k. Write µ l n the l n -th roots of unity (as a G Q -module), one has lim µ l n = H ét 1 (G m, Z l ). We may restrict the Galois representation to G Qp for any prime p. The resulting map χ cyc : G Qp Z l is always still surjective. When p l, it s easy to see that the cyclotomic 6

7 character is unramified, i.e. trivial on the inertia in G Qp. When l = p, this is no longer the case, but the geometry suggests no difference for l. This amounts to the phenomenon that when l = p, there is a notion in p-adic Hodge theory that corresponds to the usual unramified property, that χ cyc : G Qp Z p is crystalline. Lemma Any compact subgroup H GL n ( Q l ) is contained in GL n (E) for some E/Q l finite. Proof. The set H is compact Hausdorff, and thus a Baire space; it is never a countable union of nowhere-dense closed subset. Let E/Q l be any finite extension. Either the intersection H GL n (E) has finite index in H, so that by adding finitely many elements into E we have H GL n (E), or H GL n (E) has infinite index in H. We now assume the latter is the case for any E/Q l finite. There are countably many (why?) finite extensions of Q l. Thus H = E H GL n(e) as a countable union. However each H GL n (E) cannot contain an open subgroup and thus are nowhere-dense, a contradiction. Corollary Any l-adic Galois representation is realized on a E-vector space for some E/Q l finite. Example 2. Another difference that (the category of) l-adic Galois representations is different from (that of) Artin representations or representations of compact groups is that an l-adic Galois representation need not be semisimple. For example, let F be a non-archimedean local field with residue field k, char(k) l, and consider the composition χ : G F Ẑ Z l where the first map is the natural projection from G F to Gal( k/k) = Ẑ and the second map is the natural projection from Ẑ to Z l. Consider the representation ( ) ( ) 1 χ 1 χ(σ) ρ =, i.e. ρ(σ) =, σ G 1 1 F. It s obvious that ρ is non-trivial, yet it fits into an exact sequence of l-adic representations and thus ρ is not semisimple. 1 triv ρ triv 1. In general, for a (ρ, V ) be an l-adic Galois representation we will find a filtration 0 = V 0 V 1... V m = V of subrepresentations, so that each V i /V i 1 is an irreducible Galois m representation. The semi-simplification of V is V ss := V i /V i 1. Theorem (Jordan-Hölder) The semi-simplification V ss is unique up to isomorphism. Theorem (Brauer-Nesbitt) Two representations ρ and ρ of G K (in fact, any group) on a finite-dimensional vector space over any field have isomorphic semi-simplification if and only if ρ(σ) and ρ (σ) have the same characteristic polynomial for all σ G K. 7 i=1

8 Let (ρ, V ) be any l-adic Galois representation of G K, where by Lemma 10.2 we take V to be an E-vector space, E/Q l finite. Let Λ V be any lattice, i.e. a O E -submodule of rank equal to dim E V. Let GL(Λ ) := {g GL(V ) gλ Λ }. This is an open subgroup of GL(V ), and thus H := ρ 1 (GL(Λ )) is open. Let σ 1,..., σ s be a set of representative for G K /H. Then Λ = s i=1 ρ(σ i)λ is a ρ(g K )-stable lattice. Now let Λ be an arbitrary ρ(g K )-stable lattice of V. We may consider the quotient Λ := Λ/ϖ E Λ. We have a natural induced action of G K on Λ. Corollary The semi-simplification of the G K -representation Λ does not depends on the choice of Λ. Proof. The characteristic polynomial of any element in G K on Λ comes from reducing the characteristic polynomial on V modulo ϖ E, and thus independent of Λ. The result then follows from Theorem Example 3. Let p be a prime with l p + 1. We have a natural surjection Z p 2 F l. By taking any isomorphism F l {x Q l x l = 1} and Q p = Z 2 p 2 Z, we have a character χ : Q p Q 2 l, which extends to a 1-dimensional l-adic Galois representation χ : G Qp 2 Q l. Let ρ = indg Qp G Qp χ : G Qp GL 2 (Q l ). This is an irreducible representation, 2 while ρ ss = triv sgn Qp 2 /Q p. We now fix some E/Q l finite and a non-archimedean local field F with residual characteristic p l. Let G F I F P F be the absolute Galois group, the inertia, and the wild inertia, respectively. For an l-adic Galois representation ρ : G F GL(V ), we first look at how ρ IF looks like. Recall that we have I F /P F = ( Ẑ p ) := p p Z p. (Sketch of proof: By Hensel lemma style argument, any degree m extension of F ur with p m is of the form F ur ( m ϖ F ) for a fixed uniformizer ϖ F.) Write t l : I F Z l the composition I F I F /P F Z l. Definition Say a compact topological group G is prime to l if x x l is a homeomorphism. We have ker(t l ) is prime to l. One easily proves the following (see also exercise) Lemma If G is a prime-to-l group and ρ : G GL n (E) an l-adic representation, then the image of ρ is finite. Corollary (Grothendieck s l-adic monodromy theorem) Let ρ : G F GL n (E) be an l-adic Galois representation. Then there exists a relatively open subgroup U I F and a necessarily unique n n nilpotent matrix N M n (E) such that ρ(τ) = exp(t l (τ)n), τ U. 8

9 Proof. By Lemma 10.8, ker(ρ) contains an open subgroup of ker(t l ). By shrinking U we may assume ρ U ker(tl ) is trivial. We have seen that the image of ρ stabilize a lattice, and thus by conjugation we may assume Im(ρ) GL n (O E ). By shrinking U again we may assume ρ(u) K (2) := {g GL n (O E ) g = Id mod l 2 }, so that it makes sense to talk about log ρ(τ) for τ U. We have t l (U) log = U/(U ker(t l )) K (2) l 2 M n (O E ). Since t l (U) = l m Z l for some m, the above composition has to be given by x xn for some N M n (E). This is the N we seek. It remains to prove that N is nilpotent. Let Φ G F be any (lift of) arithmetic Frobenius. Then for any τ I, we have t l (ΦτΦ 1 ) = q F t l (τ) where q F is the order of the residue field of F. Applying this to the definition of N we see ρ(φ)nρ(φ) 1 = q F N. The eigenvalues of N thus have to be all zero and N has to be nilpotent. Example 4. We have the following l-adic Galois representation ρ : G F GL 2 (Q l ) given by ( ) χcyc t ρ = l. 0 1 This is somewhat an example of Galois representation that is not semisimple, but the ρ(φ) is semisimple, i.e. diagonalizable over Q l. It is a general conjecture that Frobenius should always acts semisimply. In fact, this is probably the most important non-trivial Galois representation; it s the Tate module of an elliptic curve that has semistable but not good reduction. We have understood the difference between local Artin representations and l-adic Galois representations of G F up to the inertia I F. To state the final result in a better way and to connect with Langlands program, we d like to switch from the absolute Galois group G F to the so-called Weil group. Recall that we have an exact sequence 1 I F G F Ẑ 1 We also suppose that our local class field theory is normalized so that an arithmetic Frobenius Φ is mapped to the inverse of a uniformizer. We then define the Weil group W F of F to be the preimage of Z Ẑ, but equipped with a new topology similar to that of Z Ẑ. That is, the topology on W F is such that I F has the same subspace topology as before but is open in W F. Note also that by local class field theory we have WF ab = F. We also denote by : W F C the normalized norm, so that Φ = q 1 F. An l-adic Weil representation for F is a homomorphism ρ : W F GL(V ). One easily sees that the result of Corollary 10.9 also holds for Weil representations. We now fix a Weil representation (ρ, V ) where V is an E-vector space. Let Φ again be any (lift of) Frobenius. We have 9

10 Theorem (Deligne) Let t l be as above and N as in Corollary The formula ρ (Φ m τ) := ρ(φ) m ρ(τ) exp( t l (τ)n), m Z, τ I F defines a new representation of W F on V. Moreover, the isomorphism class of (ρ, V ) does not depend on the choice of Φ. Proof. Recall that we had t l (ΦτΦ 1 ) = q F t l (τ) ρ(φ)nρ(φ) 1 = q F N. A similar argument gives t l (στσ 1 ) = σ t l (τ) and ρ(σ)nρ(σ) 1 = σ N for any σ W F. Similarly we have ρ(σ) exp(xn)ρ(σ) 1 = exp( σ xn). Now for the first statement of the theorem, suppose we have Φ m 1 τ 1 Φ m 2 τ 2 = Φ m 1+m 2 τ, i.e. τ = Φ m 2 τ 1 Φ m 2 τ 2. Then ρ (Φ m 1+m 2 τ) = ρ(φ m 1+m 2 τ) exp( t l (τ)n) = ρ(φ m 1 τ 1 )ρ(φ m 2 τ 2 ) exp( t l (Φ m 2 τ 1 Φ m 2 )N) exp( t l (τ 2 )N) = ρ(φ m 1 τ 1 )ρ(φ m 2 τ 2 ) exp( q m 2 F t l (τ 1 )N) exp( t l (τ 2 )N) = ρ(φ m 1 τ 1 ) exp( t l (τ 1 )N)ρ(Φ m 2 τ 2 ) exp( t l (τ 2 )N) = ρ (Φ m 1 τ 1 )ρ (Φ m 2 τ 2 ). For the second statement, suppose Φ is a different choice of Frobenius, and Φ = Φ τ, with t l (τ ) = s. Then the resulting ρ has ρ (Φ) = ρ (Φ τ ) = ρ (Φ) exp( sn). On the other hand, we have ρ(φ) exp(xn)ρ(φ) 1 = exp(xn) q F ρ(φ) = exp(xn) q F ρ(φ) exp(xn) 1 ρ(φ) exp(xn) 1 q F = exp(xn) q F ρ(φ)(exp(xn) q F ) 1. Taking x = s, we get ρ q F 1 (Φ) = ρ(φ) is conjugate to ρ (Φ) exp( sn) = ρ (Φ) by exp(xn)q F. Since for any τ I we have ρ (τ) = ρ (τ ) commutes with N and exp(xn) by the formula, we have ρ is conjugate to ρ by exp(xn). Remark ρ IF is trivial on the open subgroup U I F in Corollary 10.9, and thus has finite image and is a semisimple representation. On the other hand, in Corollary 10.9 we have N = 0 iff ρ IF has finite image, which is also equivalent to ρ = ρ. Definition A Weil-Deligne representation is a pair (ρ, N) where ρ : W F GL n (C) is a continuous homomorphism, and N M n (C) is a nilpotent n n matrix in C such that ρ (σ)nρ (σ) 1 = σ N for any σ W F. When N = 0, it s also common to call it a Weil representation. We fix an isomorphism ι : Q l C. Corollary Upon the choice of ι, Theorem gives a bijection between the set of isomorphism classes of l-adic Weil representations and the set of isomorphism classes of Weil-Deligne representations. 10

11 We can also talk about L-functions of Weil-Deligne representations. Definition Let ρ = (ρ, N) be an n-dimensional Weil-Deligne representation. Let V := ker(n) I F be the subspace of ker(n) on which ρ (I F ) acts trivially. Then L(s, ρ) := det(id V q s ρ (Φ) V ) 1. Lemma Let (ρ, N) be a Weil-Deligne representation. If ρ (Φ) is semisimple, then ρ (σ) is semisimple for any σ W F. Proof. Let U I F be as in Corollary By shrinking U we may assume U is normalized by W F. For any σ = Φ m τ W F, we my write σ a = Φ am τ a for any a Z. The image of τ a in I F /U has to be periodic, and thus τ a U for some a > 0. This gives ρ (σ) a = ρ (Φ) am is semisimple. Since C has characteristic zero, ρ (σ) is also semisimple. (If m = 0, then σ I F and the semisimplicity of ρ (σ) is automatic as ρ IF has finite image. Definition A Weil-Deligne representation is called Frobenius-semisimple if it satisfies the property in Lemma A general conjecture is that Weil-Deligne representations that correspond to interesting (i.e. those that come from geometry) l-adic Weil representations should be Frobeniussemisimple. What is very exciting is then that Frobenius-semisimple Weil-Deligne representations can be related to representation theory by the local Langlands conjecture. Remark. Theorem 10.10, Definition and all those follow also work for arbitrary reductive group. For example, let V be any quadratic space over Q. Then SO(V ) is a reductive group, and one may consider ρ : W F SO(V Q Ql ) and ρ : W F SO(V Q C). Then N should lie in the Lie algebra of SO(V Q C), namely a anti-self-adjoint nilpotent operator on V. 11