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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 $\mathrm{GL}_n$?

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The exact statement (which also Richard Borcherds mentions) can be also found in Serre, A Course in Arithmetic, Section 6.5. Serre points to Siegel's Gesammelte Abhandlungen, n° 20, for a proof. –  Lennart Meier Apr 2 '13 at 21:07

2 Answers 2

up vote 13 down vote accepted

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.

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Thanks, then my guess was wrong. Can you suggest a good reference for first reading about that stuff? –  plusepsilon.de Apr 3 '11 at 17:08

Conceivably it is useful to imbed Siegel's result in a somewhat larger context, so that it is a bit of a special case of something. Namely, (and this is still a special case...), even-sized orthogonal groups $O(Q)$ defined over a number field $k$ "pair" with $Sp(2n,k)$ for all choices of size $2n$, as mutual commutators inside a two-fold cover $Mp(2n\cdot \dim Q)$ of $Sp(2n\cdot \dim Q)$. The "Segal-Shale-Weil/oscillator" repn of the adele group (restricted from a repn of the metaplectic Mp) gives a well-defined mapping from repns (local or global) from irreducibles of $O(Q)$ to irreducibles of $Sp(2n)$ for $2n\gg \dim Q$, and in the other direction for the opposite inequality. (The precise cut-offs are about "first-occurrence", as in work of Kudla-Rallis and others.)

In a rather degenerate situation, for $\dim Q\gg 2n$, the constant $1$ on the adelic orthogonal group is mapped to some sort of automorphic form on $Sp(2n)$. It is not profoundly difficult, but non-trivial, to see that the image is the Siegel-type Eisenstein series. With sufficiently many decades of hindsight, and with the benefit of working on adele groups rather than classically, as Siegel did, various people have found simplified arguments: there is a bibliography and an example of a simplified argument on-line in http://www.math.umn.edu/~garrett/m/v/siegel_weil.pdf, also in the Eisenstein Series volume from AIM Harris-Skinner-Li have a wider-scope discussion in "A simple proof of rationality of Siegel-Weil" (http://www.math.jussier.fr/~harris/resarticles/SW.pdf)

The translation into classical language engenders the discussion of lattices in a genus and cardinalities of automorphism groups, much as comparison of idele class groups to ideal class groups and unit groups gives rise to classical details regarding the latter.

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