International Congress of Mathematicians

574 D. Gaitsgory1.3. Theorem. To every irreducible a as above there corresponds a (non-zero)function f a £ Yunct(GL n (X)\GL n (A)/GL n (G)), such thath-U = xAh) • U, Vft e H(GL„(A),GL„(O)).Moreover, such f„ is unique up to a scalar, and is cuspidal (see below).1.4. We will now sketch the construction oi /„• using the method oi Piatetski-Shapiro and Shalika, ci. [PS],[Sha].First, let us recall the notion oi cuspidality oi a iunction on GL„(A). Let / £Funct(G7/„(A)) be left-invariant with respect to a subgroup F (DC), where F c GL nis a subgroup that contains the unipotent radical N of the standard Borei subgroup.In particular, F contains also the unipotent radical N(Q) of any standard parabolic0 C GL n . The function / is called cuspidal if for every such parabolicy€N(Q)(X)\N(Q)(A)f(y -x) = 0,Vx£ GL n (A). (1.2)For fc = 1,..., n let Pu C GL n be the group of matrices, for which the ay entryis8ij if i > n — k,i > j. Note that P := Pi is the so-called mirabolic subgroup, i.e.the subgroup of matrices whose last row is (0, ...,0,1), and for any fc, Pj. D N.Yet N/, c GL n be the subgroup of (strictly) upper-triangular matrices, inwhich only the last fc — 1 columns may be non-zero. We shall fix a non-trivialcharacter tp : DC\A —t Q ( . It gives rise to a character \Pj. : A r j ! (A) —t Q ( , but takingthe sum of values of tp on the supdiagonal matrix entries.We define the space W* to consist of all functions/ G Funct(7*(DC)\GL„(A)/GL„(Q)),that satisfy f(n • x) = \Pfc(n) • f(x), Vn £ N/,. We define W cusp k as the subspaceof W* that corresponds to cuspidal functions. It is easy to see that for fc = n,W — W* v cusp n — * v n-1.5. Proposition. There are isomorphisms W cusp u — W cusp u+i f° r k = 1,..., n —1, which respect the H(GL„(A),GL n (