In "Mackey - Unitary Group Representation in Physics, Probability and Number Theory" on page 326, Mackey mentions a result of Siegel, which was later generalized to semi-simple Lie groups. 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, which is an L function and is then associated to a Theta series through its Mellin transform, 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 $\mathrm{GL}_n$?

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 $\mathrm{GL}_n$?