A remark on Whittaker functions on SL$(n,{\Bbb R})$

Ishii, Taku

doi:10.5802/aif.2104

Cited by 6 publications

(9 citation statements)

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“…In this section we solve the recurrence relation (1.1) to find an explicit formula for fundamental Whittaker function. Similar formulas for SL n (R) are given in [6] and [9]. We use the Pochhammer symbol (a…”

Section: Explicit Formulas For Fundamental Whittaker Functionsmentioning

confidence: 99%

Whittaker functions on orthogonal groups of odd degree

Ishii

2011

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We give explicit formulas for Whittaker functions for the class one principal series representations of the orthogonal groups SO2n+1(R) of odd degree. Our formulas are similar to the recursive formulas for Whittaker functions on SLn(R) given by Stade and the author [9]. Some parts of our results are announced in [8].where w 0 is the longest element in W. We refer to this Jacquet's Whittaker function as class one Whittaker function. Inspired by the work of Harish-Chandra, Hashizume [3] proved an expansion formula for the class one Whittaker function:Here c ′ (ν) is a product of c-functions and certain ratio of gamma functions and M ν,η is a power series solution (we call it fundamental Whittaker function) of the system (2) around the regular singularity. We will recall the results of Hashizume in section 1.Despite the development of the study of the expansion formulas, explicit formulas of the spherical functions or Whittaker functions themselves seem to be still missing in most cases. It is necessary to understand deeper the both sides of the expansion formula above for serious applications to automorphic forms.As for the fundamental Whittaker functions, recurrence relations characterizing the coefficients of them are easily given since they are essentially controlled by the Casimir operators. At the present these coefficients for SL 3 (R), SO 5 (R) and SL 4 (R) are known to be expressed in terms of (terminating) generalized hypergeometric series 2 F 1 (1)(=ratio of gamma functions by Gauss' formula), 3 F 2 (1) and 4 F 3 (1), respectively (see [1], [7], [14]). But such a direction, that is, unit arguments of generalized hypergeometric series do not seem be appropriate. Recently Stade and the author [9] reached a very satisfactory expression, which is a recursive relation between SL n−1 (R) and SL n (R). In section 2, a similar formula for SO 2n+1 (R) will be given.Jacquet integrals are actually integral representations of class one Whittaker functions. But it does not seem to be suitable form for applications to automorphic forms, such as computations of archimedean L-factors, or giving sharp estimates for Whittaker functions. Then we need to modify Jacquet integrals to more desirable expressions such as Mellin-Barnes type. In the case of SL 2 (R) the class one Whittaker function is essentially the modified K-Bessel function and it has the integral representations

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Section: Explicit Formulas For Fundamental Whittaker Functionsmentioning

confidence: 99%

Whittaker functions on orthogonal groups of odd degree

Ishii

2011

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“…The first of these identities may be deduced from (4), (11), and (15), and the second from (6), (9), and (19). In addition, we will encounter in Section 2.3 the Bessel function J ν (z) of the first kind, defined by…”

Section: Introductionmentioning

confidence: 96%

“…In [9,24], and [26], the authors of the present work obtained inductive formulas expressing principal series GL(n, R) Whittaker functions in terms of their counterparts on GL(n − 2, R). (The paper [9] treated fundamental Whittaker functions; in [24] the class one analogs were analyzed; in [26] the Mellin transforms of these class one functions were investigated.)…”

Section: Introductionmentioning

confidence: 98%

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New formulas for Whittaker functions on GL(n,R)

Ishii

Stade

2007

Journal of Functional Analysis

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We derive new recursive formulas for principal series Whittaker functions, of both fundamental and class one type, on GL(n, R). These formulas relate such Whittaker functions to their counterparts on GL(n − 1, R), instead of on GL(n − 2, R) as has been the case in numerous earlier studies. In the particular case n = 3, and for certain special values of the eigenvalues, our formulas are seen to resemble classical summation formulas, due to Gegenbauer, for Bessel functions. To illuminate this resemblance we also derive, in this work, formulas for certain special values of GL(3, R) fundamental Whittaker functions. These formulas may be understood as analogs of expressions derived by Bump and Friedberg for class one Whittaker functions on GL(3, R).

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“…And further investigation of the class one Whittaker functions is developed gradually by the papers of Stade and Ishii (cf. [8,9,[20][21][22]). …”

Section: Introductionmentioning

confidence: 97%