×

Test vectors for linear forms. (English) Zbl 0768.22004

In the study of automorphic representations it often happens that one wants to know whether a certain invariant form given by an integral is identically zero or not. In the paper the local question is treated in two situations. The first one concerns \(K^*\)-invariant linear forms on irreducible representations of \(G=GL(2,F)\times K^*\), where \(K\) is a quadratic extension of \(F\), the second one concerns \(GL(2,F)\)-invariant linear forms on irreducible representations of \(G = GL(2,F)^ 3\). In both cases a criterion is known for the existence of a non-trivial invariant linear form, which is then unique up to a scalar factor. In this paper it is shown that this linear form is non-zero on the line fixed by a certain open compact subgroup of \(G\). Hence any non-zero vector on that line is a test vector for an invariant linear form. Some global applications are given.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields

References:

[1] [C] Cartier, P.: Representation of ?-adic groups. In: Borel, A., Casselman, W. (eds.) Automorphic forms, representations andL-functions. (Proc. Symp. Pure Math., vol. 33, Part 1, pp. 111-155) Providence, RI: Am. Math. Soc. 1979
[2] [Ca] Casselman, W.: On some results of Atkin and Lehner. Math. Ann.201, 301-314 (1973) · Zbl 0239.10015 · doi:10.1007/BF01428197
[3] [D] Deligne, P.: Les constants des equations fonctionnelles des fonctions L. In: Deligne, P., Kuyk, W. (eds.) Modular Forms of one variable II. (Lect. Notes Math., vol. 349, 501-597) Berlin Heidelberg New York: Springer 1973
[4] [Ga] Garrett, P.: Decomposition of Eisenstein series: Rankin triple products. Ann. Math.125, 209-235 (1987) · Zbl 0625.10020 · doi:10.2307/1971310
[5] [Go] Godement, R.: Notes on Jacquet-Langlands theory. Princeton: Institute for Advanced Study 1970
[6] [Gr] Gross, B.H.: Local orders, root numbers, and modular curves. Am. J. Math.110, 1153-1182 (1988) · Zbl 0675.12011 · doi:10.2307/2374689
[7] [Gr-K] Gross, B.H., Kudla, S.: Heights and the central critical values of triple productL-functions. Compos. Math. (1991) (to appear)
[8] [H-K] Harris, M., Kudla, S.: The central critical value of a triple productL-functions. Ann. Math. (To appear)
[9] [J] Jacquet, H.: Automorphic forms on GL2, Part II. (Lect. Notes Math., vol. 278) Berlin Heidelberg New York: Springer 1972 · Zbl 0243.12005
[10] [J-L] Jacquet, H., Langlands, R.P.: Automorphic forms on GL2. (Lect. Notes Math., vol. 114) Berlin Heidelberg New York: Springer 1970
[11] [P] Prasad, D.: Trilinear forms for GL2 of a local field and ?-factors. Compos. Math.75, 1-46 (1990) · Zbl 0731.22013
[12] [PS-R] Piatetski-Shapiro, I., Rallis, S.: Rankin tripleL-functions. Compos. Math.64, 31-115 (1987) · Zbl 0637.10023
[13] [S] Serre, J.-P.: Trees. Berlin Heidelberg New York: Springer 1980
[14] [Ta] Tate, J.T.: Number theoretic background. In: Borel, A., Casselman, W. (eds.) Automorphic forms, representations andL-functions. (Proc. Symp. Pure Math., vol. 33, Part 2, pp. 3-26) Providence, RI: Am. Math. Soc. 1979
[15] [T] Tunnell, J.: Local ?-factors and characters of GL2. Am. J. Math.105, 1277-1308 (1983) · Zbl 0532.12015 · doi:10.2307/2374441
[16] [W] Waldspurger, J.-L.: Sur les valeurs de certaines fonctionsL automorphes en leur centre de sym?trie. Compos. Math.54, 173-242 (1985) · Zbl 0567.10021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.