3
votes
1answer
330 views

What sort of ring-theoretic properties does the representation ring of a compact Lie group possess?

Recall the definition of the representation ring $R(G)$ of a compact Lie group $G$. I'd like a reference that gives me basic ring-theoretic properties that $R(G)$ always has, or enough info that I can ...
6
votes
1answer
318 views

Representation ring and induced representation

Let $i:H \to G$ be a homomorphism of compact Lie groups. The induced representation $\iota_*V := \mathrm{Map}^H(G,V)$ of an $H$-representation $V$ does not give an element of the representation ring ...
1
vote
0answers
148 views

Equivariant $K$-theory, singular vectors, and flag manifolds

For a homogeneous space $M = G/B$, with $G$ a (complex) semi-simple Lie group, it is very well-known that equivariant vector bundles $E$ over $M$ correspond to representations $(V_{\lambda},\lambda)$ ...
1
vote
0answers
113 views

Cotorsion theory and its relative homology

Let (F(R), Cot(R)) be a cotorsion theory, Such that F(R) is the class of flat R-modules and Cot(R) the cotorsion modules. Why this is true that, For $ N\in Cot(R) $, $ \text{Ext}_{F(R)}^i(M, N)\cong ...
0
votes
1answer
173 views

About regular local rings and Socles

Let R be a regular local ring with $ \text{dim} R = d $. If $ 0\rightarrow R\rightarrow I_0\rightarrow ...\rightarrow I_d\rightarrow 0 $. Then why for $ 0\leq i\leq d-1 $, the socle of $ I_i $ is ...
0
votes
1answer
231 views

Length of a module

Let R be a commutative ring, M an R-module of finite length and let N be an Injective R-module with zero socle. Then why $ \text{Hom}_R(M, N) $ is zero?
2
votes
1answer
269 views

Kasparov's Dirac element and the index map

In Kasparov's 1988 paper Equivariant KK-theory and the Novikov conjecture section 4 he defined the Dirac element for a (non-spin) $G$- Riemanian manifold $X$ as an element in the $K$-homology ...
6
votes
0answers
534 views

Is there any “deep” relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory

First let's consider equivariant cohomology: if a compact Lie group $G$ acts on a compact manifold $M$. We have the equivariant cohomology $ H_G(M)$ defined as the cohomology of the cochain complex ...
3
votes
0answers
168 views

The proof of the splitting principle in equivariant K-theory via flag manifolds

In Atiyah's famous paper "Bott periodicity and the index of elliptic operators" section 4, he proved the splitting principle for unitary groups (Propostion 4.9 in that paper), namely: Let $j: ...
6
votes
1answer
239 views

Projective modules over non-rational group rings

Let $G$ be a finite group. We know that the $K$-group $K_0(QG)$ of the rational group ring $QG$ is a free abelian group generated by the irreducible representations of $G$ over $Q$. Now let $R$ be a ...
2
votes
0answers
205 views

k-theory of $\mathbb{Z}$

I have a doubt. Borel computed the rank of the higher algebraic k-theory of $\mathbb{Z}$: $rank(K_n)(\mathbb{Z})= 1$ if $n\equiv1 mod4$, otherwise this rank is equal to 0. On the other hand Bjorn ...
5
votes
2answers
282 views

Induction theorems for finite-dimensional complex representations of infinite groups

Let $G$ be a group, usually infinite. I am interested in finite-dimensional complex unitary representations of $G$, i.e. group homomorphisms $G \rightarrow U_n(\mathbb{C})$. The category of these ...
3
votes
2answers
277 views

Extensions which define the same element of $\text{Ext}^n(M,N)$ are in fact equivalent

It is well known (and wouldn't be so-named unless it were) that: If $\xi$, $\eta$ are $n$-fold extensions of $N$ by $M$ (modules over a ring $R$) which yield the same element of $\text{Ext}^n(M,N)$, ...
0
votes
1answer
259 views

Can injective resolutions be 'enlarged' (or shrunk) to admit only injective maps from extensions?

Let $M$ and $N$ be $R$-modules for some ring $R$. There is a standard result involving the computation of $\text{Ext}^n(M,N)$, using projective resolutions, which says that you can always choose a ...
5
votes
1answer
313 views

Splitting principle in equivariant cohomology

The following is a weaker version of what is called splitting principle in Appendix C, page 12, see also for a lighter version Brions Eq cohom and eq intersection theory, page 6: Let $G$ be a compact ...
2
votes
2answers
230 views

Action of centralizer on Borel-Moore homology of Springer Fibers for Affine Hecke Algebra

In Chriss and Ginzburg's "Representation Theory and Complex Geometry", they describe a geometric construction of representations of the affine Hecke algebra, using the Borel-Moore homology of ...
8
votes
1answer
340 views

Heller operator without left adjoint?

Suppose given a noetherian ring $R$. On the stable category $R\text{-}\underline{\text{mod}} := R\text{-mod}/R\text{-proj}$, we have the Heller operator $$ \Omega : R\text{-}\underline{\text{mod}} ...