Does anyone know the classification of irreducible admissible representations of GL(n) (over real,complex and p-adic fields), or some references?
Sorry if this question is not appropriate here.
[Math] classification of irreducible admissible representations of GL(n)
rt.representation-theory
Related Solutions
The following answer is for finite groups.
In characteristic zero, the group algebra is semisimple, so there are finitely many simple representations. These representations correspond to the blocks in the decomposition as a product of matrix algebras.
Here is a way to determine the number: Start with the field $K$, adjoin the $g$th roots of unity ($g = |G|$) to get $L$, and consider the Galois group $\Gamma_K=L/K$. This is a subgroup of the multiplicative group of the integers mod $g$. Then $\sigma_t \in \Gamma_K$ corresponding to $t \in (\mathbb{Z}/g\mathbb{Z})^*$ acts on $G$ by raising $x \in G$ to the $t$-th power. The dimension of the space of class functions constant on $\Gamma_K$-orbits is the number of simple $K$-representations.
As for characteristic p, this is modular representation theory, The number of irreducibles is the number of $p$-regular conjugacy classes (where $p$-regular means the period is prime to $p$), when the field contains the $g$th roots of unity for $g$ the order of the group. See, e.g., Serre's Linear Representations of FInite Groups. My guess is that it should be true even without the assumption on the field being sufficiently large.
Concerning
are the centers of the division rings Galois over $k$
A finite dim'l $k$-algebra $A$ is split provided $\operatorname{End}_A(S) = k$ for every simple $A$-module $S$.
[This terminology is consistent with that used in other contexts -- $A$ is split just in case the reductive quotient of the "unit group" $A^\times$ is a split reductive algebraic group over $k$.]
Suppose that $k$ is perfect and that $A_\ell = A \otimes_k \ell$ is a split $\ell$-algebra for a finite, separable extension $\ell \supset k$. Then we have the following:
$(*)$ If $S$ is a simple $A$-module, the center $Z$ of the division $k$-algebra $\operatorname{End}_A(S)$ is a subfield of $\ell$.
To see this, I claim first that $\operatorname{End}_A(S) \otimes_k \ell$ is a split semisimple $\ell$-algebra. Indeed, since $k \subset \ell$ is separable, the $A_\ell$-module $S \otimes_k \ell = S_\ell$ is semisimple, say $S_\ell = \bigoplus_i S_i$ as $A_\ell$-module, where $S_i$ is the $T_i$-isotypic component of $S_\ell$ and where $T_i$ are distinct simple $A_\ell$-modules. Since by assumption $\operatorname{End}_{A_\ell}(T_i) = \ell$, we see that $$\operatorname{End}_A(S) \otimes_k \ell \simeq \operatorname{End}_{A_\ell}(S_\ell) = \prod_i \operatorname{End}_{A_\ell}(S_i)$$ is a product of full matrix algebras over $\ell$. Now observe that the center of a split semisimple $\ell$-algebra is a a split commutative etale $\ell$-algbra $\ell \times \cdots \times \ell$. Assertion $(*)$ now follows.
Apply this to $A = kG$ for a finite group $G$. By Torsten's comment (following Emerton's answer) we may suppose $k$ to be perfect. Then $A \otimes_k \ell$ is split for an Abelian extension $\ell$ of $k$ -- a suitable $\ell$ can be obtained by adjoining to $k$ enough roots of unity. It follows that $Z$ is always Galois over $k$ (for $A = kG$).
Best Answer
There's a classification of irreducible admissible representations of real and complex reductive algebraic groups (in particular, GL(n)) due to Langlands, which is the basis of the local Langlands conjecture. A reference for this material is Knapp's "Representation theory of semisimple groups: an overview based on examples".
For GL(n) over a nonarchimedean local field, the papers "Induced reprsentations of reductive p-adic groups I,II" of Bernstein and Zelevinsky provide one step of the classification. It leaves the classification of the supercuspidal representations undertermined. Bushnell and Kutzko's "The admissible dual of GL(N) via compact open subgroups" determines the full set of irreducible admissible representations.
Update: I second Buzzard's comment above, especially with regards to the "Motives" proceedings. Kudla's and Knapp's articles in Motives II are quite nice, and contain several references including the ones I've mentioned.