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I won't say anything here about the number of components in your restricted representation, but here is some information about multiplicities.
Kwangho Choiy and Dipendra Prasad have (independently) ...

Two bad principles that taste worse together: Decimals are the true numbers. Rounding makes no difference.
Since students learn about decimals after they've learned about whole numbers and fractions,...

Here is a proof in the case where $p \not |m$ that doesn't involve modular representations, though it does involve a result of Gelfand and Kazhdan whose proof I can't remember, so I have no idea ...

An earlier answer tells you what happens when $n=3$. This is a special case of the Heisenberg group (at least in odd characteristic; not sure otherwise), any exposition of which might be illuminating ...

The set of equivalence classes of irreducible, smooth representations of a reductive $p$-adic group $G$ should be partitioned into finite subsets called $L$-packets. Each $L$-packet should correspond ...

May I assume that $K/k$ is separable?
Let $T$ be a maximal torus in $G$. Since $G$ is simply connected, the weight lattice and character lattice for $T$ are the same. This remains true if we ...

Whether breadth is considered to be negative or positive will depend on the kind of job for which you are applying.
For example, consider a high-powered research department looking for someone in ...

Here is how the real and p-adic situations are the same.
Let $G$ be a connected reductive algebraic group defined over a field $F$ not of characteristic two. Let $\theta$ be an involution of $G$ ...

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