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Frobenii on Morava $E$-theoretical quantum groups

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 Added by Gufang Zhao
 Publication date 2021
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and research's language is English




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In this paper, we study a family of new quantum groups labelled by a prime number $p$ and a natural number $n$ constructed using the Morava $E$-theories. We define the quantum Frobenius homomorphisms among these quantum groups. This is a geometric generalization of Lusztigs quantum Frobenius from the quantum groups at a root of unity to the enveloping algebras. The main ingredient in constructing these Frobenii is the transchromatic character map of Hopkins, Kuhn, Ravenal, and Stapleton. As an application, we prove a Steinberg-type formula for irreducible representations of these quantum groups. Consequently, we prove that, in type $A$ the characters of certain irreducible representations of these quantum groups satisfy the formulas introduced by Lusztig in 2015.



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Let $A$ be a finite abelian $p$ group of rank at least $2$. We show that $E^0(BA)/I_{tr}$, the quotient of the Morava $E$-cohomology of $A$ by the ideal generated by the image of the transfers along all proper subgroups, contains $p$-torsion. The proof makes use of transchromatic character theory.
The Morava stabilizer groups play a dominating role in chromatic stable ho-motopy theory. In fact, for suitable spectra X, for example all finite spectra, thechromatic homotopy type of X at chromatic level n textgreater{} 0 and a given prime p islargely controlled by the continuous cohomology of a certain p-adic Lie group Gn,in stable homotopy theory known under the name of Morava stabilizer group oflevel n at p, with coefficients in the corresponding Morava module (En)$star$X.
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