Black walnut can create a toxic area of soil beneath the trees. Its not advisable to plant anything that would even be outside the drip line of black walnut trees. Suggest that you choose a low shrub ground cover that does best in mostly shade, like Periwinkle (Vinca).

Not being specific about lavender, as mentioned by posters BW is kind of toxic to certain plants. This tixicity is due to an oil that exist in BW, which works like weed killewr. Even the tree's own seed will not thrive near it.
Most damaging effect comes from the roots of BW, making the soil intolerable to certain plants. I know for fact that tomatos, potatoes will not thrive if planted in the soil contaminated by BW's leaves, roots and wash out from its branchs. My tomato plants did not survive last year. But I had planted potatoes in containers(with soil from somewher else) did survive. Of course, there are a lot of plants that are immune to the oil in BW.I had a good crop of sesame leaves at the same area that tomatoes died.

I have neighbors with 4 huge BW trees and nothing grows under them. Lavender does like full sun.

Could you start cuttings from the old plants and have new plants?

Has anyone thought of making a weed killer from BW nut shells? Wonder if it'd kill poison ivy.

An herbalist friend grows hostas under her black walnuts. She says her hostas thrive. But lavender? Nah, they like more sun than they would get under a walnut tree.

FataMorgana

It is not just the shade of BW tree, There is more.
BW tree releases a chemical, which I think is called Joglanole. This chemical exists in all part of the tree from root, to bark, and leave.
Additionally, the root line might be even bigger than its canapy.
I have read an illinois univrsity study about this, which has listed plants that are compatible and others that cannot thrive. Over the years, BW would contaminate the soil to the extent that you cannot change it. And there is no chemical to nullify that joglanole. Last year I planted tomato in a soils brough from another location. Tomato plants weredoing fine. But when thir roots reached the existing soil, they wilted and died'
But there are a lot of things that are gowing just fine under BW tree, Automn olive, hawthorn, creeping myrtle (vinca?), something from hosta family, honeysucle,..But maple, pine, BW own seeding will not survive.

i planted lavender under black walnut tree, not thriving, need to move it