Symplectic Steinberg group

Question

I have several questions about Steinberg group and K₂ for symplectic group:

1. Can I extend the definition of Steinberg symbols to symplectic case? Will they generate the center of Steinberg group?
2. Does the center of symplectic Steinberg group coincide with K₂ (the kernel of $\mathrm{SpSt}\rightarrow\mathrm{Sp}$ as usual)?
3. Is there an analogue for Matsumoto's theorem?

I tryed to read "Sur les sous-groupes arithmetiques des groupes semi-simples deployes" by Hideya Matsumoto and all I got to know about symplectic case is that there is some problems with long roots in C_l. Also, is it written in english anywhere about non-A_l K-theory? There is "The Classical groups and K-theory" by Hahn and O'Meara, but it tells about SL and about unitary groups only.

Answer 1

There is useful information about the symplectic analogues of the Steinberg group, the Steinberg symbols, and the $K_2$ functor in some of Michael Stein's papers from the '70's. In particular, see his papers "Generators, relations and coverings of Chevalley groups over commutative rings" and "Surjective stability in dimension $0$ for $K_{2}$ and related functors" and "Injective stability for $K_{2}$ of local rings".

Answer 2

Here's a small follow-up on Matsumoto's thesis, which deals essentially with the congruence subgroup problem for Chevalley (split) algebraic groups. This followed work by Bass-Milnor-Serre, but in turn was followed by more technical work on nonsplit groups (Prasad-Raghunathan, in particular). In my 1980 Springer Lecture Notes 789 on Arithmetic Groups, I tried in the last part to convey Matsumoto's ideas in the special case of $SL_n$ when $n \geq 3$ but with some side remarks about the general case. The congruence subgroup problem has a different solution for $n=2$, which should for this and some other purposes be assigned to type $C_1$ rather than the conventional $A_1$. For symplectic groups there is a significant difference, mentioned in my Remark on page 129. Briefly:

The ideas in my $SL_n$ proof carry over almost unchanged to other Chevalley groups of rank $\geq 2$, with one important modification due to the fact that real Lie groups of type $C$ starting with rank 1 have an infinite cyclic fundamental group while others have a finite fundamental group. This is what complicates the proof for symplectic groups in Matsumoto's paper (which is only available in French). The special feature of root systems of type $C$ seems to be the existence of roots equal to twice a weight.

Answer 3

Let me do a bit of necromancy here and address the third question.

A Note on Milnor–Witt K-Theory and a Theorem of Suslin by K. Hutchinson and L. Tao provides a description for $H_2\left(Sp(F),\mathbb{Z}\right)=H_2\left(SL(2,F),\mathbb{Z}\right)$ for an infinite field $F$ as Milnor—Witt K-theory $K_2^{MW}(F)$, introduced by F. Morel in 2003 in his study of $\mathbb{A}^1$-homotopy theory.

$K_*^{MW}(F)$ is a graded associative ring generated by the symbols $[u]$, $u\in F^*$ of degree $+1$ and one symbol $\eta$ of degree $-1$ modulo the following relations:

• For $a\in F\setminus\{0,1\}$, $[a]\cdot[1-a]=0$;
• For $a,b,\in F^*$, $[ab]=[a]+[b]+\eta[a][b]$;
• For $u\in F^*$, $[u]\eta=\eta[u]$;
• $\eta^2[-1]+2\eta=0$.

The proof is based on Matsumoto—Moore presentation for $H_2\left(Sp(F),\mathbb{Z}\right)$ and the coincidence of $K_2^{MW}(F)$ with $K_2^{MM}(F)$.

PS. The equality $H_2\left(Sp(F),\mathbb{Z}\right)=H_2\left(SL(2,F),\mathbb{Z}\right)$ has something to do with $\mathsf{A}_1=\mathsf{C}_1$ (see this MO question).

Answer 4

There is an excellent survey paper: Linear Algebraic Groups and K-theory http://users.ictp.it/~pub_off/lectures/lns023/Rehmann/Rehmann.pdf by Ulf Rehmann. It seems that Matsumoto paper concerns symplectic case, that is the answers to all your questions are positive.

Note that for non-symplectic groups, Steinberg symbols are bilinear, however for symplectic it is not true. For a nice description of the 2-cocycle of the topological universal cover $\widetilde{SL_2({\mathbb R})}$ of $SL_2({\mathbb R})$ see e.g. Asai, T.: The reciprocity of Dedekind sums and the factor set for the universal covering group of $SL(2,{\mathbb R})$.

From Asai work, one can deduce that the Steinberg Symbol corresponding to a $\widetilde{SL_2({\mathbb R})}$ is defined as: for $x,y\in {\mathbb R}^{\times}$

$c(x,y) = \left\{ \begin{array}{l l} -1 & \quad \text{if } x < 0 \text{ and } y<0 \\ 0 & \quad \text{otherwise}\\ \end{array} \right.$

Answer 5

Recently, centrality and Suslin's local-global principle for symplectic K_2 are studied by Andrei Lavrenov, student of Nikolai Vavilov.