On the schwartz space of the basic affine space

Alexander Braverman, David Kazhdan

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13 Citations (Scopus)


Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G/U where U is the group of fc-points of a maximal unipotent subgroup of G. In this paper we construct a certain canonical G-invariant space S(X) (called the Schwartz space of X) of functions on X, which is an extension of the space of smooth compactly supported functions on X. We show that the space of all elements of S(X)I, which are invariant under the Iwahori subgroup / of G, coincides with the space generated by the elements of the so called periodic Lusztig basis, introduced recently by G. Lusztig (cf. [10] and [11]). We also give an interpretation of this space in terms of a certain equivariant K-group (this was also done by G. Lusztig - cf. [12]). Finally we present a global analogue of S(X), which allows us to give a somewhat non-traditional treatment of the theory of the principal Eisenstein series.

Original languageEnglish
Pages (from-to)1-28
Number of pages28
JournalSelecta Mathematica, New Series
Issue number1
Publication statusPublished - 1999
Externally publishedYes


  • Automorphic forms
  • Representation theory


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