# Does Deligne's exceptional series lead to an “exceptional K-theory”?

To a certain extent, Deligne's exceptional series $$A_1 \subset A_2 \subset G_2 \subset D_4 \subset F_4 \subset E_6 \subset E_7 \subset E_8$$ plays a role analogous to the classical series $$A_n \subset A_{n+1} \subset \dots \subset SL(\infty)$$, $$C_n \subset C_{n+1} \subset \dots \subset Sp(\infty)$$, and $$D_n \subset B_n \subset D_{n+1} \subset \dots \subset Spin(\infty)$$.

One place the classical series occur is in constructions of K-theory. In particular, it is approximately true that if you apply the Quillen plus construction to $$BSU(\infty)$$, you get complex K-theory, and if you look instead at $$\mathbb{F}$$-points of $$GL(\infty)$$, you get algebraic K-theory of the field $$\mathbb{F}$$. I imagine that similar constructions starting with $$Spin$$ and $$Sp$$ give versions of real K-theory, but I don't know.

Question: What is the result of Quillen's plus construction applied to $$BE_8$$? Does it produce an "exceptional K-theory"?

• What's the analog of $E(R)$? – AHusain Oct 12 '18 at 2:14
• I don't understand the question. Isn't $E_8$ connected, so that its classifying space is simply connected? (There's more than one thing you could mean by $E_8$, but aren't the real and complex forms are all connected?) Then what could it mean to apply the plus construction? Note, as in @AHusain's comment, that you need to specify a perfect normal subgroup of $\pi_1$ in order to speak about the plus construction. For the same reason, I'm confused when you talk about applying the plus construction to $BSU(\infty)$, which again is a simply connected space. – Dan Ramras Oct 12 '18 at 4:42
• @DanRamras Right. I was imprecise in part because of a lack of knowledge (and in part because I haven't thought through things). I could mean to take the discrete group of points of E8 over some field. That group is usually simple... – Theo Johnson-Freyd Oct 12 '18 at 12:11
• @SebastianGoette Perhaps. IIRC, Deligne's suggestion (which I think is known not to work) was that there might be a continuous family of symmetric monoidal categories, analogous to his $GL_t$, in which these eight entries were the "group" ones. (Incidentally, I think the groups he used were $G = Aut(\mathfrak{g})$ for the Lie algebras listed. So for example $D_4$ really means $PSO(8) \rtimes S_3$.) The point was that these eight groups have some modules that fuse in some particular way. – Theo Johnson-Freyd Oct 21 '18 at 23:48
• For me the interest was that, for the simply-connected groups with those names, these inclusions are all "level one", akin to the classical series $SL_n \subset SL_{n+1}$ and $Sp_n \subset Sp_{n+1}$ and, for $n\geq 5$, $Spin_n \subset Spin_{n+1}$. By "level one" I mean that the induced maps on $H^4(BG)$ are isomorphisms. So it is a sort of "homological stability", which is important in the K-theory case. – Theo Johnson-Freyd Oct 21 '18 at 23:52