recollection - transchromatic character theory

Context: Algebraic Loop Spaces and Spherical Derived Traces

Idea. Let $E$ be a Morava E-theory of height $n$. (Lubin Tate-theory from some field $l$) Its chromatic height zero localization (i.e. rationalization) $L_{K(0)}{E}_n$ can be used to capture higher-height data as follows.

Let $G\curvearrowright X$ be a finite $G$-space, and $G$ is a finite group.

Consider the following two $\infty$-groupoids associated with this action.

• $X/G$ the anima of orbits
• $I^n(X,G)$ the “n-fold inertia space”: $I^n(X,G):=\left({⨆}_{\alpha \in {\mathbb{Z}}_p^n\to G}{X}^{\alpha }\right)/G$ Notation: from now on we’ll assume $G$ is a $p$-group.

Example In the case $n=1$, the disjoint union is indexed over p-elements of $G$. Hence, the formula for $I^1(X,G)$ reads like a categorifiation of (the p-typical) Burnside’s formula

$$|X//G|=\left(\sum _{g\in G}|X^g|\right)/|G|$$

This formula can be seen as counting cells in the quotient space via the Baez-Dolan cardinality. More generally, the $n$-th inertia space $I^n(X,G)$ can be thought of groupoids categorifying height-(n+1) cardinality.

For examples, see (Elliptic III, 3.4, Formal Loop Spaces)

Diving from height n to height 0 (HKR)

Character theory is a relationship

• height $n$ cohomlogy of the $X/G$ $⇝$ height 0 cohomology of $I^n(X,G)$

where we’re thinking specifically about the cohomology theory $E_n$. In formulas, it’s a comparison

$E^{\bullet }(X/G)\to (L_{K(0)}E{)}^{\bullet }(I^n(X,G))$ This is an equivalence after rationalizing both sides.

Diving from height n to height m (Stapleton)

This relationship has a tower of generalizations. There’s a sequence of spectra interpolating between the spectra appearing on the two sides of the HKR-formula

$L_{K(n)}E,L_{K(n-1)}E,\cdots L_{K(0)}E$ Stapleton’s character formula is a comparison $E^{\bullet }(X/G)\to (L_{K(n-k)}E{)}^{\bullet }(I^k(X,G))$

We can imagine this arising from a tower

$E^{\bullet }(X/G)=(L_{n}E{)}^{\bullet }(I^0(X,G))\to (L_{n-1}E{)}^{\bullet }(I(X,G))\to (L_{n-2}E{)}^{\bullet }(I^2(X,G))\to \cdots$

This a sequence of exactly n-arrows, terminating at the final term

$\cdots \to (L_{0}E{)}^{\bullet }(I^n(X,G))$ Our notation is that $L_n$ is localizing into the corresponding thick subcategory, and we’ve written $I^0(X,G):=X/G$.

This looks like it clearly wants to be understood as adjunctions along the categorification tower.

Remark Elliptic-III’s approach to this question is by comparing tempered local systems on $I^0(X,G)$ to those on $I^n(X,G)$. It’s clear that a more direct approach with explicit examples connecting to categorification should be possible.,