Representations with Iwahori-fixed vectors Paul Garrett garrett/ 1. Generic algebras

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1 (February 19, 2005) Representations with Iwahori-fixed vectors Paul Garrett garrett/ Generic algebras Strict Iwahori-Hecke algebras Representations with Iwahori-fixed vectors Proof of the Borel-Matsumoto theorem Irreducibility criteria Using the ideas of [Casselman 1980] descended from the Borel-Matsumoto theorem on admissible representations of p-adic reductive groups containing Iwahori-fixed vectors, it is possible to give an easily verifiable sufficient criterion for irreducibility of degenerate principal series. This result is not comparable to irreducibility results such as [Muić-Shahidi 1998], but is easily proven and easily applied. Let G be a p-adic reductive group, P a minimal parabolic, N its unipotent radical, B the Iwahori subgroup matching P, and K a maximal compact subgroup containing B. As usual, a character χ : P/N C is unramified if it is trivial on P K. Let δ = δ P be the modular function on P, and ρ = ρ P = δ 1/2 P the square root of this modular function. 1. Generic algebras Let (W, S) be a Coxeter system, and fix a commutative ring R. We consider S-tuples of pairs (a s, b s ) of elements of R, subject to the requirement that if s 1 = ws 2 w 1 for w W and s 1, s 2 S, then a s1 = a s2 and b s1 = b s2. Refer to the constants a s, b s as structure constants. Let A be a free R-module with R-basis {T w : w W }. Theorem: Given a Coxeter system (W, S) and structure constants a s, b s (s S) there is exactly one associative algebra structure on A so that T s T w = T sw for l(sw) > l(w) T 2 s = a s T s + b s T 1 for all s S and with the requirement that T 1 is the identity in A. With this associative algebra structure, we also have T s T w = a s T w + b s T sw for l(sw) < l(w) Further, the right-handed version of these identities hold T w T s = T ws for l(ws) > l(w) T w T s = a s T w + b s T ws for l(ws) < l(w) Granting the theorem, for given structure constants define the generic algebra A = A(W, S, {(a s, b s ) : s S}) to be the associative R-algebra determined by the theorem. Remark: If all a s = 0 and b s = 1 then the associated generic algebra is the group algebra of the group W over the ring R. When (W, S) is affine suitable structure constants yield the Iwahori-Hecke algebra of associated p-adic groups. Most often, this is a s = q 1 b s = q 1

2 where q is the residue field cardinality of a discrete valuation ring. Proof: First, we see that the right-handed versions of the statements follow from the left-handed ones. Let l(wt) > l(w) for w W and t S. Since l(wt) > l(w), there is s S such that l(sw) < l(w). Then l(sw) + 1 = l(w) = l(wt) 1 l(swt) = l(swt) l(sw) + 1 Thus, l(w) = l(swt) > l(sw) Then T w = T s T sw and T w T t = (T s T sw )T t = T s T swt = T wt where the first equality follows from l(w) = l(ssw) > l(sw), the second follows by induction on length, and the third follows from l(sswt) > l(swt). This gives the desired result. If l(wt) < l(w), then by the result just proven T wt T t = T w. Multiplying both sides by T t on the right yields T w T t = T wt T 2 t = T wt (a t T t + b t T 1 ) = a t T wt T t + b t T wt = a t T w + b t T wt where we computed T 2 t by the defining relation. Thus, the right-handed versions follow from the left-handed. Next, suppose that l(sw) < l(w) and prove that T s T w = a s T w + b s T sw If l(w) = 1, then w = s, and the desired equality is the assumed equality T 2 s = a s T s + b s T 1 Generally, from l(s(sw)) = l(w) > l(sw) follows T s T sw = T w. Then T s T w = T 2 s T sw = (a s T s + b s T 1 )T sw = a s T s T sw + b s T sw = a s T w + b s T sw as asserted. Thus, the multiplication rule for l(sw) < l(w) follows from the rule for l(sw) > l(w) and from the formula for T 2 s. Uniqueness is easy. If w = s 1... s n is reduced, then T w = T s1... T sn Therefore, A is generated as an R-algebra by the T s (for s S) and by T 1. The relations yield the rule for multiplication of any two T w1 and T w2. Now prove existence of this associative algebra for given data. Let A denote the free R-module on elements T w for w W. In the ring End R (A) we have slleft multiplications λ s and right multiplications ρ s for s S given by λ s (T w ) = T sw for l(sw) > l(w) λ s (T w ) = a s T w + b s T sw for l(sw) < l(w) ρ s (T w ) = T ws for l(ws) > l(w) ρ s (T w ) = a s T w + b s T ws for l(ws) < l(w) Grant for the moment that the λ s commute with the ρ t. Let λ be the subalgebra of E generated by the λ s. Let ϕ : λ A by ϕ(λ) = λ(t 1 ). Thus, for example, ϕ(1) = T 1 and, for all s S, ϕ(λ s ) = T s. Certainly ϕ is a surjective R-module map, since for every reduced expression w = s 1... s n ϕ(λ s1... λ sn ) = (λ s1... λ sn )(1) = λ s1... λ sn 1 T sn = λ s1... λ sn 2 T sn 1s n =... = T s1...s n = T w 2

3 To prove injectivity of ϕ, suppose ϕ(λ) = 0. We will prove, by induction on l(w), that λ(t w ) = 0 for all w W. By definition, ϕ(λ) = 0 means λ(t 1 ) = 0. Now suppose l(w) > 0. Then there is t S so that l(wt) < l(w). We are assuming that we already know that ρ t commutes with λ, so by induction on length. λ(t w ) = λ(t (wt)t ) = λρ t T wt = ρ t λt wt = 0 Thus, λ is a free R-module with basis {λ w : w W }. This R-module isomorphism also implies that λ w = λ s1... λ sn for any reduced expression w = s 1... s n. The natural R-algebra structure on λ can be transported to A, leaving only the checking of the relations. To check the relations suppose that l(sw) > l(w). For a reduced expression w = s 1... s n the expression ss 1... s n is a reduced expression for sw. Thus, That is, we have the desired relation λ s λ w = λ sw. λ s λ w = λ s λ s1... λ sn = λ sw We check the other relation λ 2 s = a s λ s + b s λ 1 by evaluating at T w A. For l(sw) > l(w), λ 2 s(t w ) = λ s (λ s T w ) = λ s (T sw ) = a s T sw + b s T w = If l(sw) < l(w), then = a s λ s T w + b s λ 1 T w = (a s λ s + b s λ 1 )T w λ 2 s(t w ) = λ s (λ s T w ) = λ s (a s T w + b s T sw ) = This proves that λ 2 s = a s λ s + b s λ 1, as desired. = a s λ s T w + b s T s T sw = a s λ s T w + b s λ 1 T w = (a s λ s + b s λ 1 )T w The argument is complete except for the fact that the left and right multiplication operators λ s and ρ t commute with each other. A little exercise on Coxeter groups prepares for this. Proposition: Let (W, S) be a Coxeter system, w W, and s, t S. If both l(swt) = l(w) and l(sw) = l(wt), then swt = w (and s = wtw 1.) In particular, a s = a t and b s = b t, since s and t are conjugate. Proof: Let w = s 1... s n be a reduced expression. On one hand, for l(sw) > l(w) l(w) = l(s(wt)) < l(sw) so the Exchange Condition applies. Namely, there is v W such that sw = vt and such that either v = ss 1... ŝ i... s n or v = w. But v = ss 1... ŝ i... s n is not possible, since this would imply that l(wt) = l(s 1... ŝ i... s n ) < l(w) contradicting the present hypothesis l(wt) = l(sw) > l(w) On the other hand, for l(sw) < l(w) = l(s(sw)), the hypotheses are met by sw in place of w, so the previous argument applies. Thus s(sw) = (sw)t, which gives w = swt. Now the commutativity of the operators λ s and ρ t. Lemma: For s, t S, the operators λ s, ρ t commute. Proof: Prove that λ s ρ t ρ t λ s = 0 by evaluating the left-hand side on T w. There are few possibilities for the relative lengths of w, sw, wt, swt, and in each case the result follows by direct computation, although we need to use the proposition from above in two of them: 3

4 If l(w) < l(wt) = l(sw) < l(swt), then by the definitions of the operators λ s, ρ t we have In the opposite case l(w) > l(wt) = l(sw) > l(swt), λ s ρ t T w = λ s T wt = T swt λ s ρ t T w = λ s (a t T w + b t T wt ) = a t (a s T w + b s T sw ) + b t (a s T wt + b s T swt ) which, by rearranging and reversing the argument with s and t interchanged and left and right interchanged, is a s (a t T w + b t T wt ) + b s (a t T sw + b t T swt ) = ρ t λ s T w In the case that l(wt) = l(sw) < l(swt) = l(w) invoke the proposition just above. We have a s = a t and b s = b t and sw = wt. Then compute directly as desired. When l(wt) < l(w) = l(swt) < l(sw) λ s ρ t T w = λ s (a t T w + b t T wt ) = a t (a s T w + b s T sw ) + b t T swt = a s (a t T w + b t T wt ) + b s T swt = ρ t (a s T w + b s T sw ) = ρ t λ s T w λ s ρ t T w = λ s (a t T w + b t T wt ) = a t T sw + b t T swt = ρ t (λ s T w ) A corresponding argument applies in the case opposite to the previous one wherein l(sw) < l(w) = l(swt) < l(wt). When l(w) = l(swt) < l(sw) = l(wt) again invoke the proposition above to obtain a s = a t and b s = b t and also sw = wt. Then λ s ρ t T w = λ s T wt = a s T wt + b s T swt = a t T sw + b t T swt = ρ t T sw = ρ t λ s T w This finishes the proof of commutativity, and of the theorem on generic algebras. 2. Strict Iwahori-Hecke algebras This section demonstrates that Iwahori-Hecke algebras attached to groups acting suitably on buildings are generic algebras in the sense above. The argument depends only upon the local finiteness of the building. Let G be a group acting strongly transitively on a thick building X, preserving a labelling. (The strong transitivity means that G is transitive upon pairs C A where C is a chamber in an apartment A in the given apartment system.) Let (W, S) be the Coxeter system associated to the apartments: each apartment is isomorphic to the Coxeter complex of (W, S). Conversely, a choice of apartment and chamber within it specifies (W, S). We assume always that S is finite. The subgroup B is the stabilizer of C. The local finiteness hypothesis is that we assume that for all s S the cardinality q s = card(bsb/b) = card(b\bsb) is finite. Recall that the subgroup of G stabilizing the facet F s of C of type {s} for s S is P = P s = B{1, s}b = B BsB The subgroup B is the subgroup of P additionally stabilizing C, so BsB is the subset of B s B mapping C to another chamber s-adjacent to C (that is, with common facet F s of type {s}.) Therefore, BsB/B is in bijection with the set of chambers s-adjacent to C (other than C itself) by g gc. 4

5 That is, the local finiteness hypothesis is that every facet is the facet of only finitely-many chambers. Equivalently, since S is finite, each chamber is adjacent to only finitely-many other chambers. Fix a field k of characteristic zero. Let H = H k (G, B) be the Iwahori-Hecke algebra in G over the field k, that is, the collection of left and right B-invariant k-valued functions on G supported on finitely-many cosets Bg in G. The convolution product on H is (f ϕ)(g) = h B\G f(gh 1 )ϕ(h) The hypothesis that ϕ is supported on finitely-many cosets Bx implies that the sum in the previous expression is finite. Since ϕ is left B-invariant and f is right B-invariant the summands are constant on cosets Bg, so summing over B\G makes sense. Nevertheless, we must prove that the product is again in H. We do this in the course of the theorem. Generally, let ch E be the characteristic function of a subset E of G. By the Bruhat-Tits decomposition, if indeed they are in H(G, B), the functions ch BwB form a k-basis for H(G, B). This Hecke algebra is visibly a free k-module. Theorem: Each BgB is a finite union of cosets Bx, the algebra H is closed under convolution products, and ch BsB ch BwB = ch BswB for l(sw) > l(w) with ch BsB ch BsB = a s ch BsB + b s ch B a s = q s 1 and b s = q s That is, these Iwahori-Hecke operators form a generic algebra with the indicated structure constants. Further, for a reduced expression w = s 1... s n (that is, with n = l(w) and all s i S), we have q w = q s1... q sn Proof: First prove that double cosets BwB are finite unions of cosets Bx at the same time that we study one of the requisite identities for the generic algebra structure. This also will prove that H is closed under convolution products. Do induction on the length of w W. Take s S so that l(sw) > l(w). At g G where ch BsB ch BwB does not vanish, there is h G so that ch BsB (gh 1 )ch BwB (h) 0. For such h, we have gh 1 BsB and h BwB. Thus, by the Bruhat cell multiplication rules, g = (gh 1 )h BsB BwB = BswB Since this convolution product is left and right B-invariant ch BsB ch BwB = c ch BswB for some positive rational number c = c(s, w). Compute the constant c = c(s, w) by summing the previous equality over B\G c q sw = c card(b\bswb) = c ch BswB (g) = g B\G = c (ch BsB ch BwB )(g) = ch BsB (gh 1 )ch BwB (h) = g B\G g B\G h B\G 5

6 = ch BsB (g)ch BwB (h) = q s q w (the latter by replacing g by gh, interchanging order of summation.) Thus, c = q s q w /q sw and for l(sw) > l(w) ch BsB ch BwB = q s q w q 1 sw ch BswB This shows incidentally that the cardinality q sw of B\BwB is finite for all w W, and therefore that the Hecke algebra is closed under convolution. Now consider the other identity required of a generic algebra. Since BsB BsB = B BsB we need evaluate (T s T s )(g) only at g = 1 and g = s. For g = 1 the sum defining the convolution is For g = s (ch BsB ch BsB )(1) = h B\G ch BsB (h 1 )ch BsB (h) = q s = = (q s 1) 0 + q s 1 = (q s 1)ch BsB (1) + q s ch B (1) (ch BsB ch BsB )(s) = h B\G = card(b\(bsb BsBs)) ch BsB (sh 1 )ch BsB (h) = Let P be the parabolic subgroup P = B BsB. This is the stabilizer of the facet F s. The innocent fact that P is a group gives BsB BsBs = (P B) (P B)s = (P B) (P s Bs) = = (P B) (P Bs) = P (B Bs) Therefore, BsB BsBs consists of [P : B] 2 left B-cosets. This number is (q s + 1) 2 = q s 1. Thus, Therefore, with T w = q 1 w b s = q 1 s ch BsB ch BsB = (q s 1)ch BsB + q s ch B ch BwB we obtain a generic algebra with structure constants a s = (1 qs 1. However, this is a weaker conclusion than desired, as we wish to prove that for l(sw) > l(w) If so, then our earlier computation would show that q s q w = q sw ) and ch BsB ch BwB = ch BswB Then taking simply T w = ch BwB would yield a generic algebra with structure constants a s = q s 1 and b s = q s. On one hand, (with l(sw) > l(w)) evaluate both sides of at sw. The left-hand side is h B\G ch BsB ch BwB = q s q w q 1 sw ch BswB ch BsB (swh 1 )ch BwB (h) = card(b\(bsb(sw) BwB)) = 6

7 = card(b\(bsbs BwBw 1 )) card(b\(bss Bww 1 )) = card(b\b) = 1 The right-hand side is q s q w q 1 sw, so q s q w q sw On the other hand, invoking the theorem on generic algebras (with l(sw) > l(w)) This gives qs 1 ch BsB qsw 1 ch BswB = (1 qs 1 Evaluate both sides at w. The right side is q sw q 1 w )q 1 sw ch BswB + qs 1 qw 1 ch BwB ch BsB ch BswB = (q s 1)ch BswB + q sw qw 1 ch BwB and the left is card(b\(bsbw BswB)) = card(b\(bsb BswBw 1 )) = = card(b\(bsb BsBBwB w 1 )) card(b\(bsb BsBww 1 )) = card(b\bsb) = q s by the cell multiplication rules. That is, q sw q s q w Combining these two computations yields q sw = q s q w. Induction on length gives the assertion q s1...s n = q s1... q sn for a reduced expression s 1... s n W. Thus, we obtain the generic algebra as claimed. 3. Representations with Iwahori-fixed vectors Here we prove the Borel-Matsumoto theorem. The structure of the Iwahori-Hecke algebras is the essential ingredient in this proof. Again, G is a p-adic reductive group and B an Iwahori subgroup. More specifically, let G o be the label-preserving subgroup of G, and take B to be the subgroup in G o stabilizing a chosen chamber in a chosen apartment in the associated (affine) building. Theorem: (Borel, Matsumoto, Casselman) Let G be a reductive p-adic group with Iwahori subgroup B (unique up to conjugation) and corresponding minimal parabolic subgroup P. Let M o = B M be a maximal compact subgroup of a chosen Levi component M = M P of P. Let π be a smooth representation of G with dim π B < (Admissibility of π would assure the latter condition.) Under the quotient map q : π π N from π to its Jacquet module π N with respect to P, the B-fixed vectors π B in π map complex-linear isomorphically to the M o -fixed vectors (π N ) Mo in π N. Proof: (See [Casselman 1980]) Let P be a minimal parabolic subgroup of G matching B. We grant the general fact that for any parabolic Q and matching parahoric B Q, the B Q -fixed vectors surject to the B Q - fixed vectors in the Q-Jacquet module. Thus, the strength of the present assertion is the injectivity. Let π(n) be the kernel of the quotient mapping to the P -Jacquet module. If v π B π(n), then there is a large-enough compact open subgroup N 1 of N such that N 1 n v dn = 0 Take a M P such that an 1 a 1 N o, where B has Iwahori factorization B = N opp 1 M o N o. (The notation is potentially misleading: N 1 is large while N opp 1 is relatively small.) Further, we may take a to lie inside the label-preserving subgroup G o of G. Then 0 = a 0 = anv dn = ana 1 av dn = n av dn N 1 N 1 an 1a 1 7

8 Then Paul Garrett: Representations with Iwahori-fixed vectors (February 19, 2005) n av dn = N o N o/an 1a 1 Normalize the measure of B to be 1. Then 0 = 0 dm dn opp = M o N opp 1 an 1a 1 N o/an 1a 1 n o n av dn dn o = n o 0 dn o = 0 N opp 1 M o n opp m n av dn dm dn opp = N o B b av db Further, since v π B, Thus, from just above. That is, BaB b v db = v B x v dx = ba v db db = B B B ch BaB v = 0 ba v db = 0 For some reflections s i there is a reduced expression s 1... s n such that BaB = Bs 1... s n B By the structural results for the Iwahori-Hecke algebra, ch BaB = ch Bs1B... ch BsnB Each of the functions ch BsiB acting on π stabilizes the finite-dimensional complex vector space π B. For s S, the structural assertion ch 2 BsB = q s ch BsB + (q s 1) ch B says that any eigenvalue λ of ch BsB on π B must satisfy so λ = q s ± q 2 s 4(q s 1) 2 λ 2 = q s λ + (q s 1) = q s ± (q s 2) 2 = 1, q s 1 That is, no eigenvalue can be 0. Thus, each ch BsB gives an invertible operator on π B, and ch BaB is necessarily invertible, contradicting our earlier conclusion unless v = 0. That is, π B π(n) = 0 after all. 4. Imbeddings to unramified principal series The Borel-Matsumoto theorem on representations with Iwahori-fixed vectors gives several important corollaries about admissible spherical representations and the structure of unramified principal series. Corollary: An irreducible admissible representation π with non-zero B-fixed vector imbeds into an unramified principal series representation. Proof: Since the set π B of B-fixed vectors in π is non-trivial, by the theorem the space of M o -fixed vectors π Mo N in the Jacquet module π N is non-trivial. Since the Jacquet-module functor preserves admissibility, 8

9 π Mo N is finite-dimensional. Since P is minimal, M/M o is an abelian group. Thus, π Mo N has an irreducible P -quotient χ which is a complex one-dimensional space upon which M/M o acts. That is, M acts on χ by an unramified character (that is, a character trivial on M o ). By Frobenius Reciprocity, we obtain a non-zero G-map π Ind G P χ, which must be an isomorphism since π is irreducible. Corollary: Every G-subrepresentation and every G-quotient of an unramified principal series representation of G is generated by its B-fixed vectors. Proof: Let π be a subrepresentation of an unramified principal series I χ = Ind G P ρ χ where ρ is the square root of the modular function for P. From Frobenius Reciprocity 0 Hom G (π, I χ ) Hom M (π N, ρχ) Since π Mo N πb as complex vector spaces (from above), π B 0. That is, any subrepresentation π of an unramified principal series contains a non-zero B-fixed vector. Taking contragredients, the inclusion give rise to a surjection π I χ I ˇχ Ǐχ ˇπ Let H be the kernel. We have shown that H B is non-trivial. Corollary: If an unramified principal series representation I χ = i P χ is not generated by its unique (up to constant multiples) spherical vector ϕ (for fixed choice K of maximal compact), then there is a non-zero intertwining T : I χ I χ from I χ to another unramified principal series I χ such that T ϕ = 0. Proof: (Following Casselman) Let V be the proper submodule generated by the spherical vector. Then I χ /V 0, so is still admissible (since an unramified principal series representation is admissible, and admissibility is preserved in quotients). And it is generated by its B-fixed vectors (I χ /V ) B, by the theorem, which then necessarily form a finite-dimensional subspace. Thus, I χ /V is finitely-generated, so has an irreducible quotient σ, which (by the corollary) still has a non-zero B-fixed vector. Thus, σ imbeds into an unramified principal series I χ. That is, we have a non-zero intertwining I χ I χ /V σ I χ This proves the corollary. 5. Irreducibility criteria First recall some relatively elementary standard facts about principal series representations. Let W be the (spherical) Weyl group of G, acting on P/N by conjugation, where N is the unipotent radical of the minimal parabolic P. Let W act on one-dimensional group homomorphisms χ : P/N C by (wχ)(x) = χ(w 1 xw) 9

10 As usual, say that χ is regular if wχ = χ only for w = 1. Let I χ = i P χ denote the (smooth) normalized unramified principal series I χ = i P χ = Ind G P ρ P χ Let ϕ χ be the unique K-spherical vector in I χ such that ϕ χ (1) = 1. Theorem: Let χ be an unramified character. If T : I χ I χ is a non-zero intertwining, then χ = wχ for some w W. If χ is regular, then for all w W dim Hom G (I χ, I wχ ) = 1 In particular, for fixed regular χ and w W, the non-zero map T χ,w : I χ I wχ defined (for suitable inequalities imposed upon χ to ensure convergence, and then by analytic continuation) by T χ,w f(g) = f(w 1 ng) dn is the only such intertwining, up to a constant. N w 1 Nw\N Theorem: The holomorphically parametrized family χ T χ,w (for fixed w W ) of intertwining operators I χ I wχ has a holomorphic continuation to all regular χ. The holomorphically continued intertwining is not the zero intertwining. Theorem: ([Casselman 1980]) Let χ be a regular character, let ϕ χ be the spherical vector in Ind G P χ normalized so that ϕ χ (1) = 1. There is an explicit not-identically-zero rational function λ w (χ) of χ so that T χ,w (ϕ χ ) = λ w (χ) ϕ wχ Corollary: ([Casselman 1980]) Let λ(χ) = λ wo (χ) be the function attached to the longest Weyl element w o in the spherical Weyl group. For regular characters χ, the spherical vector ϕ χ generates I χ if and only if λ(χ) 0. Proof: Let V be the subrepresentation generated by the spherical vector. If V is not all of I χ, then the quotient I χ /V is non-zero. By the corollary of the Borel-Matsumoto theorem, this quotient has a non-zero intertwining to an unramified principal series I χ. Necessarily χ = wχ for some w W. Either by looking at the rational functions λ w (χ) directly, or by realizing that in a finite Coxeter group W for any w there is w such that the longest element w o is expressible as w o = w w with Thus, and therefore l(w o ) = l(w ) + l(w) T w,χ T w,χ = T wo,χ λ wo,χ = λ w (χ ) λ w (χ) = λ w (χ ) 0 = 0 Thus, when the spherical vector fails to generate I χ, λ wo (χ) = 0. On the other hand, if λ wo (χ) = 0, the spherical vector is in the kernel of the non-zero intertwining operator T wo (χ), so the spherical vector cannot generate I χ. Remark: These intertwining operators are well understood. A conceptual proof of meromorphic continuation can be given by some form of Bernstein s continuation principle. One version is in [Garrett 1997]. 10

11 In fact, since the compositions of these intertwinings must fall back into the same class, one can make a stronger assertion about the relationships among the intertwinings and among the λ w s for w 1 w 2 = w 3, etc. Corollary: (Casselman) For regular χ, the unramified principal series I χ λ(χ) 0 and λ(χ 1 ) 0. is irreducible if and only if Proof: Suppose V is a proper G-submodule of I χ. On one hand, if V contains ϕ χ, then ϕ χ fails to generate all of I χ, so λ(χ) = 0. On the other hand, if V does not contain ϕ χ, then the (smooth) contragredient Ǐ χ I χ 1 has a proper submodule X = {x Ǐχ : x(v ) = 0} which necessarily contains the spherical vector ϕ χ 1. That is, Ǐχ is not generated by its spherical vector, so λ(χ 1 ) = 0. Now let Q be any other parabolic subgroup containing P, with unipotent radical N Q. Let σ : Q/N Q C be an unramified character. Let i Q σ = Ind G Q ρ Q σ be the normalized degenerate principal series. By restriction, such σ gives an unramified character on P and we have an injection i Q σ = Ind G Q ρ Q σ Ind G P ρ Q σ = Ind G P ρ P ρ 1 P ρ Q σ = i P (ρ 1 P ρ Q σ) The following is an obvious extension to degenerate principal series of Casselman s results for unramified principal series, though the condition we obtain ceases to be provably necessary for irreducibility. Corollary: For an unramified character σ of Q, if ρ 1 P ρ Q σ is regular, and if λ(ρ 1 P ρ Q σ) 0 and ρ Q σ 1 ) 0, then the (normalized) degenerate principal series i Q σ is irreducible. λ(ρ 1 P Proof: First, we verify that if the spherical vector ϕ σ in i Q σ generates i Q σ, and if the same is true for the contragredient ǐ Q σ i Qˇσ, then i Q σ is irreducible. Indeed, suppose that i Q σ had a proper submodule V. On one hand, if V contains the spherical vector, then the spherical vector fails to generate i Q σ. On the other hand, if V does not contain the spherical vector, then in the contragredient the submodule X = {x ǐ Q σ : x(v ) = 0} is proper and contains the spherical vector, so the spherical vector fails to generate ǐ Q σ. Now we relate this to the generation of unramified principal series by the spherical vector. From the obvious inclusion ǐ Q σ i Qˇσ i P (ρ 1 P ρ Q ˇσ) by dualizing we obtain a surjection of the contragredients, which (by choice of the normalizations) gives a surjection i P (ρ P ρ 1 Q σ) i Qσ If the spherical vector in i Q σ fails to generate i Q σ, then surely the same is true of i P (ρ P ρ 1 Q σ). The obvious parallel remark applies to the contragredient. If ρ P ρ 1 Q σ is regular, then we may invoke Casselman s criterion for generation of unramified principal series by the spherical vector. 11

12 6. Bibliography [Casselman 1980] W. Casselman, The unramified principal series of p-adic groups. I. The spherical function, Compositio Math. 40 (1980), no. 3, [Garrett 1997] P. Garrett, Euler factorization of global integrals, in Euler Products and Eisenstein Series, Proc. Symp. Pure Math. vol. 66. [Muić-Shahidi 1998] G. Muić, F. Shahidi, Irreducibility of standard representations for Iwahori-spherical representations, Math. Ann. 312 (1998), no. 1, pp