Could you suggest me a reference where the following non-holomorphic generalization of the Eisenstein series is discussed? $$ G_{k,l}(\tau,z) = \sum_{m,n} (z+m+n\tau)^{-k}(\bar z …

Here is my setting: Let $E\in\mathcal{M}_k(\Gamma_0(N))$ be an Eisenstein series (of trivial Nebentypus) that is a normalized eigenform for all the Hecke operators at level $\Gamma …

In obtaining the spectral decomposition of $L^2(\Gamma \backslash G)$ where $G=SL_2(\mathbb{R})$, and $\Gamma$ is an arithmetic subgroup (I am satisfied with $\Gamma = SL (2,\mathb …

Given $q = e^{2\pi i \tau}$ and the Eisenstein series $E_{2k}(\tau)$, i.e., $$E_2(\tau) = 1-24\sum_{n=1}^\infty \frac{n q^n}{1-q^n}$$ $$E_4(\tau) = 1+240\sum_{n=1}^\infty \frac{n …

Are there any existing numerical libraries for Eisenstein series? In particular I am interested in calculating values of parabolic Eisenstein series on $ SL(n,\mathbb Z) \setminus …

In the context of spectral decomposition of functions in $L^2(\Gamma \backslash \mathfrak{h})$, or Selberg trace formula, we come across three different types of spectrum. First o …

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 s …

I've heard that irreducible unitary representations of noncompact forms of simple Lie groups, the first example of such a group G being SL(2, R), can be completely described and th …

The Eisenstein series $$ G_{2k} = \sum_{(m,n) \neq (0,0)} \frac{1}{(m + n\tau)^{2k}} $$ are modular forms (if $k>1$) of weight $2k$ and quasi-modular if $k=1$. It is clear that giv …

It is well known that the first Kronecker limit theorem gives the Laurent expansion of the Eisenstein series $E(z,s)$ over $SL(2,Z)$ at $s=1$; see, for example, Serge Lang's book E …

The decomposition of L^2(S^2) under SO(3; R) is well-known. Focus now on the hyperbolic plane H presented as the quotient SL(2; R)/SO(2; R). It is non-compact, therefore my under …