Timeline for Why are there so few irreducible admissible representations of $\text{GL}(n,\mathbb{R})$ (up to infinitesimal equivalence)?
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| Oct 18 at 18:13 | comment | added | Will Sawin | If you think of representations as coming from characters of maximal tori, which is not completely wrong as a description of the irreducible admissible representations even in the $p$-adic case, then this is because of the fact that maximal tori of $GL_n$ split as products of maximal tori of $GL_1$ and $GL_2$. | |
| Oct 18 at 0:57 | comment | added | LSpice | (2), yes, definitely! But also the $p$-adics have lots of algebraic extensions. (I was giving a job talk once and was asked, "can you say a word about why there's such a difference between the representation theory of real and $p$-adic [general linear] groups", to which I answered, "I'll say two words: algebraic extensions.") | |
| Oct 17 at 22:04 | comment | added | Aurel | If you believe in the Langlands correspondence, this is "because" the Weil group of $\mathbb{R}$ only has 1- and 2-dimensional irreducible representations, whereas $p$-adic Weil groups have irreps of all dimensions. In other words, it is "because" $\mathbb{R}$ has a very simple Galois group, whereas $\mathbb{Q}_p$ has a complicated one. But of course this is going backwards... | |
| Oct 17 at 21:50 | history | asked | Daniel Miller | CC BY-SA 4.0 |