An analogue of Siegel’s 𝜙-operator for automorphic forms for 𝐺𝐿_{𝑛}(𝑍)

Abstract

If 5ePn is the symmetric space of n x n positive matrices, Y £ S^P" can be decomposed into n o\Ar> o \/i rx, \x I) \ 0 vl/("-l)Wj \0 I J ' where W £ SffPn-X. By letting v-► oo we obtain the ^-operator that attaches to every automorphic form for GL"(Z), f(Y), an automorphic form for GL"_!(Z), mW). This paper can be followed with either choice of T". If y £ r" and Y £âa" , F" acts on £Pn by sending Y to Ty Yy, where Ty is the transpose. We will use Siegel's notation Y[y] for Ty Yy. An automorphic form for T" is a function f :£?"-> C satisfying the following conditions: (i) / is an eigenfunction for all the C7L"(R)-invariant differential operators. (ii) f(Y[y]) = f(Y) for all Y £ &n and y £ Yn. (hi) If Ps(Y) is Selberg's power function defined by p~s(Y) = JJ"=X \Yj\ sj for s £ C" then there are C > 0 and s £ C" such that \f(Y)\ < C\p-s(Y)I as the upper left determinants \Yj\-» 00.