Lee, Kyu-Hwan; Lenart, Cristian; Liu, Dongwen Whittaker functions and Demazure characters. (English) Zbl 1472.11162 J. Inst. Math. Jussieu 18, No. 4, 759-781 (2019). Summary: In this paper, we consider how to express an Iwahori-Whittaker function through Demazure characters. Under some interesting combinatorial conditions, we obtain an explicit formula and thereby a generalization of the Casselman-Shalika formula. Under the same conditions, we compute the transition matrix between two natural bases for the space of Iwahori fixed vectors of an induced representation of a \(p\)-adic group; this corrects a result of D. Bump and M. Nakasuji [Can. J. Math. 63, No. 6, 1238–1253 (2011; Zbl 1230.22009)]. Cited in 2 Documents MSC: 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E50 Representations of Lie and linear algebraic groups over local fields 20F55 Reflection and Coxeter groups (group-theoretic aspects) Keywords:Iwahori-Whittaker functions; Casselman-Shalika formula; Demazure characters; Demazure-Lusztig operators; shellability Citations:Zbl 1230.22009 PDFBibTeX XMLCite \textit{K.-H. Lee} et al., J. Inst. Math. Jussieu 18, No. 4, 759--781 (2019; Zbl 1472.11162) Full Text: DOI arXiv References: [1] Billey, S. and Lakshmibai, V., Singular Loci of Schubert Varieties, , (Birkhäuser Boston Inc., Boston, MA, 2000). · Zbl 0959.14032 [2] Björner, A. and Wachs, M., Bruhat order of Coxeter groups and shellability, Adv. Math.43 (1982), 87-100. · Zbl 0481.06002 [3] Brubaker, B., Bump, D. and Licata, A., Whittaker functions and Demazure operators, J. Number Theory146 (2015), 41-68. · Zbl 1366.22008 [4] Bump, D. and Nakasuji, M., Casselman’s basis of Iwahori vectors and the Bruhat order, Canad. J. Math.63 (2011), 1238-1253. · Zbl 1230.22009 [5] Casselman, W., The unramified principal series of p-adic groups I. The spherical function, Compos. Math., 40, 387-406 (1980) · Zbl 0472.22004 [6] Casselman, W. and Shalika, J., The unramified principal series of p-adic groups II. The Whittaker function, Compos. Math.41 (1980), 207-231. · Zbl 0472.22005 [7] Deodhar, V. V., Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Möbius function, Invent. Math., 39, 187-198 (1977) · Zbl 0333.20041 [8] Deodhar, V. V., A combinatorial setting for questions in Kazhdan-Lusztig theory, Geom. Dedicata, 36, 95-119 (1990) · Zbl 0716.17015 [9] Lenart, C. and Zainoulline, K., Towards generalized cohomology Schubert calculus via formal root polynomials, Math. Res. Lett. (2014), Preprint, arXiv:1408.5952 (to appear). · Zbl 1393.14048 [10] Nakasuji, M. and Naruse, H., Yang-Baxter basis of Hecke algebra and Casselman’s problem (extended abstract), in Proceedings of FPSAC 2016, , pp. 935-946 (Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2016). · Zbl 1435.20011 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.