Let $\Gamma$ be a compact Hausdorff second countable topological group. ``second countable ``the topology of $\Gamma$ has a countable base" Irr($\Gamma$) denotes the set of equivalence classes of irreducible representations of $\Gamma$. The vector space for each irreducible representation of $\Gamma$ is a finite dimensional vector space over the complex numbers. Irr($\Gamma$) is a countable set --- and we give it the discrete topology.
Let $X$ be a locally compact Hausdorff topological space with a given (continuous) action of $\Gamma$ on $X$.
For each $x$ in $X$, $\Gamma}_{x$ denotes the isotropy group of $x$. If $x$ and $y$ are in the same orbit for the given action of $\Gamma$ on $X$ , then Irr($\Gamma}_{x$) and Irr($\Gamma}_{y$) are canonically in bijection. The usual quotient, denoted $X/G\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}a$, is obtained by collapsing each orbit (for the given action of $\Gamma$ on $X$) to a point. The extended quotient, denoted $X//\Gamma$, is constructed by replacing the orbit of $x$ (for the given action of $\Gamma$ on $X$) by Irr($\Gamma}_{x$). This talk will describe how extended quotient enters into the representation theory of Lie groups and reductive p-adic groups. For Lie groups this is due to Mackey-Higson-Afgoustidis. For reductive p-adic groups, this is due to Aubert-Baum-Plymen-Soll
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The Lefschetz fixed point theorem gives an invariant that is zero when an endomorphism is homotopic to a map with no fixed points. The converse is not true unless we refine the Lefschetz number to a richer invariant called the Reidemeister trace.
When you come to these invariants as a homotopy theorist (of a particular variety) you notice that the Reidemeister trace is a topological Hochschild homology invariant and is additive. This suggests that you should look for connections to algebraic K-theory since it is the universal home for additive invariants. In this talk I'll (start to) explain the connection between traces defined from fixed point invariants and traces defined for computing K-theory