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I'm not totally sure what will satisfy you on this account. Of course you pointedly restricted yourself to the reals, but some complex manifolds do have holomorphic half-form bundles, and the flag ma …

If you identify the Hecke algebra with the Grothendieck principal block of a graded lift of category $\mathcal{O}$ for the corresponding semi-simple Lie algebra (oddly, enough which of the two Langlan …

I don't have a full answer, or a good reference, but here's how I think one should answer the question: By Jacobson-Morozov (see Chriss & Ginzburg for more details), classifying nilpotents in a semi- …

This is all on the wikipedia page for Deligne-Lusztig theory. I don't recognize the definition that you give, and can't see why it should be equivalent to the one I know. I'm not sure it's wrong, si …

I don't have a complete answer or a reference, but I have a principle: (*) a conjugacy class in GL(n) is the same as an isomorphism class of n-dimensional representation of $\mathbb{Z}$. Similarly, …

Yes. These can be proven by some very simple principles. The smallest representation of a division ring is the ring itself (this covers the first 3), since it has no left or right ideals. When yo …

Well, one "general" theorem is If A and B are semi-simple algebras acting faithfully on a finite dimensional vector space $V$ such that $\mathrm{End}_A(V)=B$ and vice versa, then tensoring with $ …

The short answer is no. You just have to think of them as formal sums, in the same way that you can only think of elements of a group algebra as formal sums. What you can do is think of the path cat …

The isomorphism works over Z. The proof is that the basis change between W-symmetrized monomials and characters is upper-triangular with 1's on the diagonal.

Certainly, there's no reason to believe there will be. The Springer correspondence depends very strongly on the existence and structure on the Springer resolution.

From looking at the slides, it sure looks like you wrote it in your answer: take 2-quotients. I'm not sure if there's a standard reference for n-quotients of partitions, but they're described in this …

Certainly it doesn't generalize in a really obvious way. One way to think about this is the following: Gabriel's theorem uses in really deep way that the category of quiver representations over a fi …

Not an answer, but here's a rephrasing: this question is equivalent to asking why parabolic restriction doesn't depend on the parabolic (parabolic restriction is the adjoint, so you restrict to the pa …

I second the suggestion of Fulton and Harris. It's a funny book, and definitely you want to keep going after you finish it, but it's a good introduction to the basic ideas. You specifically might be …

I would rather write $\chi_{R_1}(a)=\chi_{R_1^*}(a^{-1})$. This allows us to see the sum $$\frac{1}{n!}\sum_{a\in S_n}\chi_{R_1^*}(a^{-1})\chi_{R_2}(ba)$$ as the trace on $\mathrm{Hom}_{\mathbb{C}}(R …