In "Mackey - Unitary Group Representation in Physics, Probability and Number Theory" on page 326, George Mackey mentions a result of Ludwig Siegel, which was later generalized to semi-simple Lie groups by André Weil. In his words:


"Now in one formulation Siegel's main result takes the form of an identity between an theta series and an Eisenstein series."


My first guess is that this relation goes through the constant term of the Eisenstein series, which is an L function. This L function is the Mellin transform of a Theta series, right?


Q1: What is the exact statement?


Q2: Is there a nice result for $\mathrm{GL}\_2$ (in terms for intertwiner for the parabolic induced representation) for these kind of results?


Q3: How does this generalize to $\mathrm{GL}\_n$?



Relation between Theta series and Eisensteinseries
Siegel showed that an Eisenstein series is a certain constant times the sum of weighted theta functions of all lattices in some genus. The lattices are weighted by 1/automorphism group. For example, for even unimodular lattices of dimension 8 there is only one such lattice, so Siegel's result says the Eisenstein series E4 is the theta function of this lattice. 


Siegel's result does not generalize to GLn in an obvious way, but does generalize to Siegel modular forms for the symplectic group Sp2n, where the corrresponding Eisenstein series is again a linear combination of theta functions (of several variables). 


Even the special case of the trivial group Sp0 is interesting: it is the Smith-Minkowski-Siegel mass formula giving the sum of the weights of the lattices in a genus.


