This is a sequel of Yuri Berestâ€™s talk. Simple examples of derived representation schemes exhibit phenomena that are only visible in computer experiments and are only partly understood mathematically. I will discuss some of these examples, especially the derived schemes of the polynomial algebra in two variables. They are derived versions of the commuting schemes, parametrizing pairs of commuting matrices. I will introduce the Harish-Chandra homomorphism for derived representation schemes and conjecture that it is an isomorphism in the case of commuting schemes. Comparing Euler characteristics implies seemingly new combinatorial identities for matrix integrals. Generalizations to arbitrary reductive groups will also be presented. (Joint work with Y. Berest, A. Patotski, A. Ramadoss and T. Willwacher)

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