TY - JOUR
T1 - On the schwartz space of the basic affine space
AU - Braverman, Alexander
AU - Kazhdan, David
PY - 1999
Y1 - 1999
N2 - 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.
AB - 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.
KW - Automorphic forms
KW - Representation theory
UR - http://www.scopus.com/inward/record.url?scp=0013168227&partnerID=8YFLogxK
U2 - 10.1007/s000290050041
DO - 10.1007/s000290050041
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AN - SCOPUS:0013168227
SN - 1022-1824
VL - 5
SP - 1
EP - 28
JO - Selecta Mathematica, New Series
JF - Selecta Mathematica, New Series
IS - 1
ER -