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Register a new userSpecial classes of homomorphisms between generalized Verma modules for ${mathcal U}_q(su(n,n))$
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Authors
Hans Plesner Jakobsen
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We study homomorphisms between quantized generalized Verma modules $M(V_{Lambda})stackrel{phi_{Lambda,Lambda_1}}{rightarrow}M(V_{Lambda_1})$ for ${mathcal U}_q(su(n,n))$. There is a natural notion of degree for such maps, and if the map is of degree $k$, we write $phi^k_{Lambda,Lambda_1}$. We examine when one can have a series of such homomorphisms $phi^1_{Lambda_{n-1},Lambda_{n}} circ phi^1_{Lambda_{n-2}, Lambda_{n-1}} circcdotscirc phi^1_{Lambda,Lambda_1} = textrm{Det}_q$, where $textrm{Det}_q$ denotes the map $M(V_{Lambda}) i prightarrow textrm{Det}_qcdot pin M(V_{Lambda_n})$. If, classically, $su(n,n)^{mathbb C}={mathfrak p}^-oplus(su(n)oplus su(n)oplus {mathbb C})oplus {mathfrak p}^+$, then $Lambda = (Lambda_L,Lambda_R,lambda)$ and $Lambda_n =(Lambda_L,Lambda_R,lambda+2)$. The answer is then that $Lambda$ must be one-sided in the sense that either $Lambda_L=0$ or $Lambda_R=0$ (non-exclusively). There are further demands on $lambda$ if we insist on ${mathcal U}_q({mathfrak g}^{mathbb C})$ homomorphisms. However, it is also interesting to loosen this to considering only ${mathcal U}^-_q({mathfrak g}^{mathbb C})$ homomorphisms, in which case the conditions on $lambda$ disappear. By duality, there result have implications on covariant quantized differential operators. We finish by giving an explicit, though sketched, determination of the full set of ${mathcal U}_q({mathfrak g}^{mathbb C})$ homomorphisms $phi^1_{Lambda,Lambda_1}$.
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A recent result of N. Abe implies that the Gabber-Joseph conjecture is true for the first-degree extensions between Verma modules with regular integral highest weights.
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