The Casselman-Shalika Formula for a Distinguished Model

The Casselman-Shalika ormula for a Distinguished Model by William D. Banks Abstract. Unramified Whittaker functions and their analogues occur naturally in number theory as terms in the ourier expansions of automorphic forms. Precise information about these functions is useful in many aspects of study, such as in the construction of L-functions. In this paper, the method of Casselman-Shalika is used to derive explicit values for the analogue of the unramified Whittaker function in a distinguished model that arises in connection with quadratic base change.. Introduction The unramified Whittaker functions and their analogues play an important role in modern number theory, arising naturally as terms in the ourier expansions of automorphic forms. It is generally desirable whenever possible to calculate explicit values for these functions, as the information proves useful in many aspects of study related to the automorphic form for example, in the construction of associated L-functions. When an automorphic representation possesses a Whittaker model or another suitable unique model, the method described in [3] may be used to compute an explicit formula the Casselman- Shalika formula for the values of the unramified Whittaker function or the analogous function. In this paper, we consider the following distinguished model that arises in connection 99 Mathematics Subject Classification Numbers: 70, 2235

with quadratic base change and the theory of the Asai L-function see [], [5], and [6]. The existence and uniqueness of this model was established by Hakim in [4]. Let be a nonarchimedean local field of characteristic zero and with odd residue characteristic, and let q denote the cardinality of the residue field. Let / be an unramified quadratic extension of. Let O denote the ring of integers in. Put G GL2,, G GL2,, and K GL2, O. If π, V is an irreducible, admissible, unramified principal series representation of G that is trivial on the center Z of G, it is known that: i There exists a nonzero spherical vector φ 0 V, unique up to complex constant, such that πkφ 0 φ 0 for all k K. ii There exists a nonzero linear functional T : V C, unique up to complex constant, such that T πgφ Tφ for all g G and φ V see [4]. As π is in the principal series, φ 0 may be regarded as complex-valued function on G. In 6, it is shown that Tφ 0 0. Thus, we can normalize the constants in i and ii so that φ 0 on K, and Tφ 0. Now let Q be the vector space of all locally-constant complex-valued functions on G that satisfy fgg fg for all g G and g G. The group G acts on Q by right translation. Let Qπ denote the invariant subspace of Q spanned by functions Q φ of the form Q φ g Tπgφ. The map φ Q φ yields an isomorphism of G -modules V Qπ, and Qπ is called the distinguished model for V. or the above model, the correct analogue of the unramified Whittaker function is the function Q 0 Q φ 0 given by Q 0 g Tπgφ 0 for all g G. Observe that Q 0 is a well-defined function on the double cosets of G \G /K, hence it suffices to compute Q 0 2

on the complete set of double coset representatives { M k : k 0,, 2,... } described in 3. The main result of this paper proved in 6 is the following: Theorem. Let t χ GL2, C be the Satake parameter of π, and let k be the k-th symmetric tensor representation of GL2, C for each k 0. Then: and if k 2: Q 0 M 0, Q 0 M q q q Trt χ q +, Q 0 M k q k q q Tr k t χ q + Tr k t χ + Tr k 2 t χ. The author would like to thank Daniel Bump for many valuable discussions and helpful suggestions, and the referee for his careful review and insightful comments on the text. 2. Some Notation Let v be a fixed valuation of which restricts to a valuation of, and fix once and for all a prime element in such that v this is possible since / is unramified. Let O denote the ring of integers of, its unique maximal ideal, and O its group of units. Let q be the odd cardinality of the residue field O /, and let q v be the absolute value symbol for corresponding to v. We similarly define O,, O, q, and q v. Then q q 2, and 2. As / is quadratic, µ for some µ O O with µ2 O. Then µ and O O µo. By Hensel s lemma, the image of τ µ 2 in the residue field O / is a quadratic nonresidue. 3

or G GL2, let P be the standard Borel subgroup of G, let N be the unipotent radical of P, and let Z be the center of G. The Weyl group W of G consists of the elements µ {e, w 0 }, where e and w 0. Once and for all, fix ξ G. An additive Haar measure dx on is said to be normalized if volo. A multiplicative Haar measure d y on is said to be normalized if d y y dy for some normalized additive Haar measure dy on. By the Bruhat decomposition, G P P w 0 P. Then Z \P has measure 0 in Z \G, and the matrices: { } y x x w 0 : x, x and y form a complete set of distinct representatives for Z \P w 0 P. A left Haar measure dg is said to be normalized if: φg dg Z \G φ y x x dx d y w 0 y d x for all φ integrable on Z \G, where dx and d x are normalized additive Haar measures for, and d y is a normalized multiplicative Haar measure for. 3. Double Coset Decomposition The function Q 0 described in is well-defined on G \G /K, since: Q 0 gg k Tπgg kφ 0 Tπgπg πkφ 0 Tπg φ 0 Q 0 g for all g G, g G and k K. Hence, it suffices to determine Q 0 on a complete set of double coset representatives for G \G /K. 4

By the Iwasawa decomposition, a complete set of double coset representatives for { } k N \G /K is given by the matrices A : k, l Z. Then N A is a complete set of representatives for G /K. As N A AN and A G, it follows that N is a complete set of representatives for G \G /K. To reduce the set further, note that each z may be written in the form z x + u k µ, where x, u O, and k Z. Then: l z u k x k µ u, u k x u k µ with G and K. Moreover, since G K { } k µ whenever k 0, it follows that the matrices M k : k Z, k 0 form a complete set of double coset representatives for G \G /K. Moreover, it may be shown that these matrices represent distinct cosets. 4. The Linear unctional T As a principal series representation, the space V V χ of π is the vector space Ind G P δ 2 χ consisting of all locally-constant complex-valued functions on G that satisfy φpg δ 2 χpφg for all p P and g G. Here χ and δ are characters of P : y χ y 2 y δ y 2 χ y χ 2 y 2, y y 2 q vy 2 vy for all y, y 2 and, where χ, χ 2 are characters of. Since π is unramified, χ is unramified, and we can assume that χ y α vy, χ 2 y β vy for some α, β C 5

and all y. Moreover, the condition that π is trivial on Z implies αβ. Thus: y φ g βq y vy 2 vy φg 2 for all y, y 2,, g G, and φ V χ. An unramified character χ of P as above is said to dominant if β >. Proposition. Suppose that χ is a dominant character and dg is a normalized left Haar measure on Z \G. Then the integral: Tφ Z \G φw 0 ξg dg is well-defined and absolutely convergent for all φ V χ, and therefore defines a linear functional T : Vχ C such that Tπgφ Tφ for all g G and φ V χ. Proof: The only issue here is that of convergence. By uniqueness of Haar measure, the integral Tφ is a nonzero multiple of the integral: K φ w 0 ξ y x dx d y k y dk, where dk is a left Haar measure on K GL2, O, dx is a normalized additive Haar measure for, and d y is a normalized multiplicative Haar measure for. or each k K and φ V χ, let: G φ k φ w 0 ξ y x dx d y k. y Then each G φ is locally-constant on K since φ is locally-constant. As K is compact, the absolute convergence of Tφ follows from the absolute convergence of the integrals G φ k. 6

Moreover, there is no loss of generality in assuming that k e, since G φ k G πkφ e. Thus: G φ e {0} φ w 0 ξ φ w 0 ξ {0} {0} {0} φ w 0 ξ y x dx d y y y x y dx d y y x dx dy βq vy xy + yµ φ w 0 dx dy β vy x + yµ φ w 0 dx dy β vy x + yµ φ w 0 dy dx by ubini s theorem. Since χ is dominant β >, we can add the point y 0 to each inner integral to obtain: G φ e β vy x + yµ φ w 0 dy dx β v Imz z φ w 0 dz, where Imz y if z x + yµ, and dz dy dx is a normalized additive Haar measure for z. Since φ is locally constant, we can choose L 0 so that φ w 0 φw 0 if vz L, and φ z φe if vz L. Also whenever z 0 we have: z z z φ w 0 φ z z βq 2vz φ z. It follows that G φ e can be expressed as a sum of three integrals: φw 0 β v Imz dz + φe β v Imz βq 2vz dz vz L vz L 7

+ β v Imz z φ w 0 dz. L<vz<L The first two integrals converge absolutely by a straightforward calculation whenever β >, and the last integral converges absolutely for all β since it is the integral of a continuous function over a compact region. or dominant characters χ, the lemma in 6 shows that Tφ 0 0, and we can define: Tφ Tφ Tφ 0 for φ V χ. Then T : V χ C is the unique linear functional that satisfies Tπgφ Tφ for all g G, φ V χ, and Tφ 0. A flat section is a collection { φ χ Ind G P δ 2 χ : χ unramified } such that φ χ K is independent of χ. It follows from the Iwasawa decomposition that every flat section {φ χ } is equicontinuous; that is, there exists an open neighborhood K φ of the identity, which independent of χ, such that φ χ is K φ -fixed for every χ. Proposition 2. Let g be a fixed element of G, and let {φ χ } be a flat section. Then the function βq 2 T πg φ χ, initially defined for dominant characters χ i.e. β >, extends to an entire function of β C. Proof: rom the proof of Proposition when χ is dominant: G πg φ χ k φ χ w 0 kg β v Imz dz + φ χ kg β v Imz βq 2vz dz + vz L χ k β v Imz z φ χ w 0 vz L χ k kg dz, L χ k<vz<l χ k 8

for each unramified character χ and k K, where L χ k 0 is chosen so that: z vz L χ k φ χ w 0 kg φ χ w 0 kg, vz L χ k φ χ z kg φ χ kg. As {φ χ } is equicontinuous, we can choose L χ k Lk independent of χ and locallyconstant on K. As K is compact, we can choose Lk L independent of k. The first two integrals in may be directly computed for dominant χ: φ χ kg φw 0 kg vz L β v Imz β v Imz βq 2 L dz φw 0 kg, 2 βq βq 2vz dz φ χ kg β L q βq 2 βq. 3 β vz L As before, the last integral in converges absolutely for all unramified characters χ and all k K and consequently defines an entire function of β. It follows that for each k K, βq β G πg φ χ k continues to an entire function of β, and as K is compact, βq β T πg φ χ has this continuation as well. or each unramified character χ, let be the unique K -fixed vector in Ind G P δ 2 χ such that on K. Then the collection { } is a flat section. The lemma in 6 shows that: for dominant χ, hence: T q 2 + βq β βq 2 Tπg φ χ βq β Tπg φ χ q 2. 9

This equation implies the proposition. 5. Iwahori ixed Vectors Let B be the Iwahori subgroup of G i.e. the inverse image of the standard Borel subgroup under the natural homomorphism K GL2, O /. Let V B χ be the space of Iwahori fixed vectors: V B χ {φ V χ : πbφ φ for all b B }. By the Iwahori factorization, each element g G may be expressed in the form g p w g b where p P, w g W, and b B. Moreover, w g is uniquely determined by g. or each w W, let: φ w,χ g { δ 2χp if g p w g b and w g w, 0 otherwise. By a theorem of Casselman see [2], the vectors {φ w,χ } w W form a basis the Casselman basis for V B χ. Let NO N K, and let dn be a left Haar measure on N such that vol NO. or each k 0, g G, the integral: k g NO k g n dn converges absolutely for all unramified χ. Note that 0. It is known that the { k} are Iwahori fixed vectors given explicitly by: k w W c w χ δ 2 w χ k φ w,χ 4 0

where c e χ, c w0 χ βq 2 β 2, and w χ is the character given by: for all y, y 2 and. w y χ χ w y w y 2 y2 Proposition 3. or dominant χ, Q 0 M 0 T 0, and if k : Proof: We compute: T k Z \G Z \G NO Z \G Z \G Q 0 M k q k w 0 ξgdg O O O k w 0 ξgn w 0 ξg w 0 ξg T k q k. y y since G for all y O and volo. u n If x u n with u O, n Z, then dn dg xµ k dy dx dg k xµ dx dg 5 G, u K, and therefore: Z \G w 0 ξg T Z \G Z \G π k xµ dg u n w 0 ξg w 0 ξg k n µ k n µ dg k n µ u dg TQ 0 M 0 if n k, TQ 0 M k n if n < k.

Thus, applying ubini s theorem to 5, it follows that: T k k n φ 0 xµ χ w 0 ξg dg dx n 0 n k q k vxn k vol n n+ Tφ 0 χ Q 0M 0 + vol n n+ Tφ 0 χ Q 0M k n T Q 0M 0 + q n0 Tφ k 0 χ n0 q n Q 0M k n The proposition now follows from this by a simple inductive argument. 6. Calculation of Tφ w,χ and Q 0 M k Lemma. or dominant χ, we have: Tφ 0 q 2 χ + βq. β Proof: The left Haar measure dg is normalized see 2 so that: T w 0ξg dg w 0 ξ Z \G x If x O, then w 0 K, and: φ 0 y x x dx d y χ w 0 ξ w 0 y On the other hand, if x O, then x K, and: φ 0 y x x dx d y χ w 0 ξ w 0 y x 2 w 0 ξ y x x dx d y w 0 y y x x x y x 2 x + y x w 0 ξ dx d y y w 0 ξ y x dx d y. y 2 x w 0 ξ d x. y x dx d y. y dx d y x y

Thus: T q min 2v x,0 d x w 0 ξ y x dx d y y + q G φ 0 e. χ Since is K -fixed, it follows from equation of 4 that: G φ 0 χ e β v Imz βq 2vz dz + β v Imz dz. vz vz 0 To evaluate these integrals, take L 0, φ in equation 2 of 4, and L, φ φ0 χ in equation 3 of 4. The lemma follows. We now turn to the calculation of Tφ w,χ for w W. or this, it will be necessary y x x to determine those g w 0 ξ w 0 in G such that w g w 0. Thus, let λ y x + µ, η λ + x yx µx 2 τ + x, where τ µ 2. Then: w 0 ξ y x x w 0 x y λ λ λ y λ η w 0. It follows that w g w 0 if and only if η O, hence if and only if yx 2 τ O and x + yxx 2 τ O. Consequently if w g w 0 : φ w0,χg βq vyλ2 βq min 2vx,0 vy. Proposition 4. or dominant χ, we have: Tφ e,χ βq β 2 β 2, and Tφ w0,χ + βq. Proof: In this proof, ubini s theorem will be applied frequently without mention. 3

We compute: Tφ w0,χ φ w0,χ w 0 ξ yx µx 2 τ + x O y x x dx d y w 0 y βq min 2vx,0 vy d y dx y We apply a change of variables y x 2 τy o, d y d y o, where y o O {0}. Because the image of τ in O / is a quadratic nonresidue see 2, it follows that vy min 2vx, 0 + vy o, and thus: Tφ w0,χ O {0} xy o + x O d x βq vy o x 2 τ y o d y o dx d x. Next, we apply a change of variables x yo x o x, dx y o dx o, where x o O : Tφ w0,χ βq vyo yo x O {0} O o x 2 τ y o 2 dx o d y o d x β vyo q vyo d x yo x O {0} O o x 2 dx o d y o τ To compute the inner integral, we apply a change of variables x y o x o +x o, d x y o d x o : d x. d x yo x o x 2 τ y o d x o x 2 o τ y o q min 2v x o,0 d x o y o +q. Then: and therefore: Tφ w0,χ + q O {0} O β vyo dx o d y o q 2 β, Tφ w0,χ Tφ w0,χ T + βq. 4

rom 4, we know that 0, thus using equation 4 we can write: c e χφ e,χ + c w0 χφ w0,χ. Applying the linear functional T, we obtain: whence the proposition follows. c e χtφ e,χ + c w0 χtφ w0,χ Tφ e,χ + + βq β 2, Proof of Theorem: By Proposition 2, each Q 0 M k regarded as a function of β continues meromorphically to the entire complex plane, with possible simple poles only at ±q. Applying the linear functional T to equation 4, we have by Proposition 4: T k c w χ δ 2 w χ k Tφ w,χ w W q k βq β k β 2 + βq 2 q k β 2 β 2 βk + βq. Substituting α β, this simplifies to: T k q k q k α k α q α β α k+ β k+ α β β q βk α β q α k β k α β q k Sk α, β q S k α, β, where S k α, β is defined for k Z by: k α i β k i if k 0, S k α, β i0 0 otherwise. Note that each S k α, β is an entire function of β. 5

Now, by Proposition 3 when χ is dominant, Q 0 M 0 T 0, and when k : Q 0 M k q T k q k q T k q T k q k q q S k α, β q + S k α, β + S k 2 α, β. Thus, Q 0 M k continues to an entire function of β for each k. Bearing in mind that the α Satake parameter for π is the matrix t χ GL2, C, then Tr β k t χ S k α, β for each k 0, whence the theorem is established. References. T. Asai, On certain Dirichlet series associated with Hilbert modular forms and Rankin s method, Math. Ann. 226 977, 8-94. 2. W. Casselman, The unramified principal series of p-adic groups I: the spherical function, Compositio Math. 40 asc. 3 980, 387-406. 3. W. Casselman and J. Shalika, The unramified principal series of p-adic groups II: the Whittaker function, Compositio Math. 4 asc. 2 980, 207-23. 4. J. Hakim, Distinguished p-adic representations, Duke Math. J. 62 No. 99, -22. 5. G. Harder, R. P. Langlands and M. Rapoport, Algebraische Zyklen auf Hilbert-Blumenthal- lächen, Crelles Journal 366 986, 53-20. 6. Y. Ye, Kloosterman integrals and base change for GL2, Crelles Journal 400 989, 57-2. Written: August 7, 99; Revised: May 7, 993 William D. Banks, Department of Mathematics, Stanford University, Stanford, CA 94305-225 6